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.
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 sword". 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.