A BB/pellet gun is a viable killing tool.

A few years ago my son drilled a gray squirrel in the back yard. Dead. Surprised me because when I was young I shot a few squirrels using a pump-up BB gun: The shot generally stunned them enough to allow a quick clunk on the head. The autopsy my son and I conducted as part of preparation for cooking revealed he hit it at the base of the skull. Severed the spine, tore into the windpipe.

Small birds and chipmunks are no trouble to kill outright.

Pigeons, on the otherhand, simply fluff up a bit. Even head shots had no effect. We were trying to rid the office building of roosting birds. After trying some nonlethal deterants, we went with a lethal plan: A pump-up pellet gun. It didn't even make them fly.

A caveat: My shots were all with a pump-up rifle, which is much more accurate than a pistol because of the longer barrel. To effect a kill with a pistol, you're going to have to get much closer.