But it sure limits your options to brain, spine, and heart. You can forget about a "boiler room" shot that only penetrates the lungs. You may be lucky, if you're a good tracker, to find where he settles down, to finish him off, with another shot. I read about a polar bear hunt, where upon examination of the bear they shot, they found a .22 bullet in the roof of its mouth that didn't penetrate to the brain. It sucked to be that Eskimo.
Well, no one here is hunting polar bears. Likely that came from a pistol as a final act of desperate defense or an attempt to dissuade the bear form attacking. Lots of trappers carry .22 pistols.
On a deer, lung shots (especially with a small entrance wound) are a bitch to track in grasslands. A larger caliber through and through can be a challenge, too, though at least there is a little more blood trail. Heart? I have seen whitetail deer go 100+ yards with its heart in three pieces (12 ga. rifled slug), although it laid down a trail a blind man could follow. The small entrance wound (and exit wound unlikely with a .22) again, makes tracking harder.
Best target with a small caliber arm is a high neck shot. Either it hits the spine and drops it right there, takes out the bleeders in the neck (a little low, but easier tracking), or it misses. Not a long distance shot, but within 50 yards, and the deer standing still, walking slowly, or feeding. Getting that opening to shoot requires either a lot of luck, or a thorough knowledge of the area, where the deer move, when they move, and the ability to stay quiet in a decent hide, much like bowhunting, and ideally at about the same range.