The way to go for hunting with an air pistol is to the PCP guns. Unfortunately, they are expensive to set up. In addition to the cost of the gun itself, from $390 (Benjamin Marauder/ 14gr .22 at 700fps) to $600 (Evanix AR6 14 gr .22 at 1000 fps), there is the added cost of a way to charge the things with air. Usually this is done with a high pressure Scuba tank ($) and/or a three stage high pressure hand pump ($190-$240).
They do shoot, though.
Another hot PCP pistol....somewhat larger....is the AirForce Talon. A .25 caliber gun. It will shoot a 43 grain pellet at 760 fps if the descriptions are correct.
A particularly nice feature of the Talon is that it will work with CO2 bulk cylinders, though a converter is needed.
"Only hunting and mountain climbing are sports. The rest are just games." - R.Ruark
NRA Life Member