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Not possible without also ejecting particulate matter. If it could be done then people would be cut into pieces by standing near a helicopter or aircraft propeller.

The only solution would be to use sand or shot blasting. Even then, if your opponent was wearing armour, the worst you are likely to do is clean it for them.

You would also require quite a lot of equipment.

It is possible to kill with air pressure, you just have to be REALLY CLOSE to do it

The only documented death I found from "air pressure" is Jon-Erik Hexum, who was bored on the set of his tv series. He had a gun loaded with blanks, and decided to play a fake game of Russian Roulette with his gun and the blank cartridges (i.e. put the gun on your temple and pull the trigger).

This is still not safe because of both the explosion created by the blank and the bits of paper and plastic moving at high speeds out of the muzzle. He died from a fractured skull and pieces of paper and plastic that were embedded in his brain.

Blanks cause injuries if put next to the skin.

Of course, millions of actors have been "shot" by blanks with no ill effects. Most on-set fatalities involved a malfunctioning or poorly maintained gun, such as Brandon Lee.

To sum up - It is completely possible to cause injury or death with air pressure, but the death zone is basically touching the other person. Anyone with a sword could simply cut off the hand holding the "air sword", so this weapon would be completely impractical.

One other thing to consider. Blanks are meant to be a safe way to fire a gun on-screen. Real bullets came first. If you have the tech to get air moving at the speed required to injure or kill, you already have the technology to make a real gun that fires real projectiles.

EDIT:

Commenters pointed out that a shock-wave from an explosion (i.e. air pressure) can kill. This was proven repeatedly on MythBusters, which also tested Hollywood cliches for survivability. Simply jumping behind a wooden table (according to the show) is enough to protect you from the blast.

Blast-waves can kill but are even more impractical than an "air cannon". Why...

1) The attacker will be at the epicenter of the blast. They will likely be more injured than the victim.

2) Simply running away or jumping behind a wooden table would protect the victim

3) (Same as blanks) If you have the tech to get here, then you can make a grenade, which is obviously deadly.

No, especially not slashing.

Punching bags can really take a beating because the sand inside absorbs the blows. The incoming blow slams straight into the side of the bag, which directly transfers that energy into the sand inside. The first layer of sand broadcasts that energy to the sand above, below, and beside. As each "layer" of sand transfers energy to the next layer, it diffuses that energy, spreading it outward from where your first originally struck. Not only that, but there are little gaps between the grains of sand, and as the sand slips and jostles, it burns the energy it was given on friction. Thus, even a strong punch will only lightly shift the sand in the bag, and even that will just shift back a second later as the bag settles.

But what does sand have to do with air?

Well, air is like the sand in the bag - except that air molecules have even more distance between them. A "sharp" blast of air will be barely a breeze a step away. Air focused in a tiny stream will quickly push outward, expanding into the lower pressure space in front of it. As it expands, it carries less and less energy - what used to be millions of molecules traveling together quickly becomes thousands, or tens. The amount of air it would take to actually hurt someone would be immense, even at shorter-than-pocket-knife range.

This is just kind of related, but it might interest you that there is a small crab or shrimp that launches sonic/shock waves with their claws: https://www.youtube.com/watch?v=XC6I8iPiHT8
(also quite cute!))

The main ingredient of this effect seems to be a delicate buildup and release of pressure, giving you an idea how envision physically sound way to weaponize "air slashes": https://en.wikipedia.org/wiki/Alpheidae#Snapping_effect

As pointed out in the adjoining replies, a blast/slash of air cannot occur on its own, from a physical viewpoint. You can either compress that air and direct it at the target and this would require insanely high precision to focus it and cause significant damage. Also, as the other reply states, "air slashes" are basically shock waves travelling in air. Now, from my knowledge of undergraduate fluid mechanics, a shock wave occurs when a fluid is made to undergo a careful transition from subsonic state to supersonic state by making it flow through a convergent-divergent nozzle. In order to create the shock wave at the exit plane of the nozzle (which literally is your weapon), you need to operate it very precisely under the "third design pressure ratio point" which, in practice, is too difficult to achieve. Also, once the shock wave exits the nozzle, it can no longer be treated as a one-dimensional wavefront, rather it becomes three-dimensional, expanding and dissipating exponentially. So, that would require your opponent to be present within a few meters of your reach.

No, its not really possible.

Sound is just a pressure wave. If you have a powerful enough sound wave, it can do some damage.

Loudest possible sound wave is 194 decibels. This is a result of the wave trough being a vacuum (and the wave crest being 2 atm.)

This sound level is considered approximately the level required to be fatal. Human studies are necessarily restricted. It is thought that this sound level could induce fatal embolisms in the lungs.

Since flesh has considerable flexibility, it is resilient against sonic assault. Something rigid like glass or concrete would be more easily damaged, and can in fact be destroyed when a frequency closely match to a natural vibration mode is used. This is not a sonic cutting attack though.

194 Db is not powerful enough to cut flesh. So, you cannot use a cutting sound strike no matter how you wave your weapon. In theory, if your weapon was large enough, and moving fast enough, it could be sufficient to cause damage due to hearing hearing loss, or even a stunning effect, but never cut flesh.

A metal cutting water jet shoots at 760 m/s you could attach one to your sword.

Speed of sound 343 m/s if your character arm
can move that fast then it would be deadly. (why not just use throwing weapon or a gun)

A wind with maximum sharpness is a discontinuity - where the velocity suddenly changes between different regions of air, rather than continuously (gradually) changing from one velocity to another.

I don't know how dangerous air shock waves are, but water shock waves can klll you underwater. Probably, a sufficiently strong air shock wave (i.e. large enough change in velocity) will be as deadly as you want.

Shockwaves are commonly spherical, heading out in all directions from a point, but I think some directionality must be possible, just as sound wave and light waves can have some directionality. Perhaps, originating from within a pipe?

However, a shock wave generated by a sword slashing (extremely quickly) would travel laterally to the sword - like the bow wave of a ship. A force-push (extremely quickly) might work better.

disclaimer all off the top of my head, not an expert

Yes, it is possible. How you you do it? You attach a shaped charge explosibe to your sword. A bomb's shock wave, when properly shaped, is certainly capable of cutting people, and the medium for the shock wave is the air between the bomb and its target.

This is very far-fetched, but at least sort-of based in real science.

A magnifying glass can concentrate sunlight enough to start a fire.

Lithotripsy is a medical procedure where shock waves are passed through the body of the patient at different angles so that they target kidney stones, causing them to break apart. The shock waves are designed to come to a focal point centered on the stones to mimimize damage to other tissue. The same concept is applied in some forms of radiation therapy.

A whispering gallery is a room or environment which takes advantage of the reflection of sound from curved walls to allow soft sounds to be heard far away. In some cases an ellipsoidal chamber is used. A very faint sound made at one focal point reflects from the curved walls and can be heard clearly at the other focal point, even though it can't be heard in other parts of the chamber.

Imagine combining these concepts in a special environment such that a series of loud sounds generated at one point echo through a chamber, and are timed so perfectly that the multiple echoes all arrive at a single point simultaneously, producing a very small but intense sound, possibly enough to cause injury.

I know this is a physics/science question, but I think the "mechanism" used in the anime situations you're describing is magic. A lot of martial arts (and especially kung fu) fiction invokes some kind of magical element, like the flying in Crouching Tiger, Hidden Dragon (2000). Kung Fu Panda 3 (2016) explains it as "ch'i" which is a concept used by real-life martial arts teachers to get students thinking about energy flowing through their bodies, but fiction usually takes the concept to a fantastically supernatural level. The idea of cutting at a distance isn't so different from a voodoo doll situation, and is seen in movies like Kung Fu Hustle (2004) by means of a magical stringed instrument rather than any kind of cutting implement. There's no reason to think the air is actually what's doing the cutting by itself.

The reason why air doesn't cut in an atmosphere, is because the atmosphere is able to absorb all of it. There's a lot of particles of nitrogen that can get hit by the 'gun' and then re-direct the energy in another direction. Sand blasters work because the sand has enough momentum to move the atmosphere out the way and keep the force directed in the right direction.

Water cutters used on earth are approximately 2k bar (30k psi) - and that pressure is able to cut sheet metal without issue - though they do cut slowly. If that pressure was applied by air, it would cut exactly the same (because the force on the metal is the same).

Commercial air pumps exist that are able to output air at this high pressure; they're not small. You'll also need to store quite a significant amount of gas to use this device too - so realistically, you're not going to have it on your person. You can however run a pipe from the pump to the weapon.

The range will still be limited as the particles in the beam will still spread out - but no where near as limited on earth; but I wouldn't want to be within a few meters of the front of this weapon. Especially as it doesn't actually need to do very much damage to someones space suit/craft to result in a leak.

Given the number of anime programs that are set in space with crafts that fight with swords, this seems like it is still on topic!