You might be able to kill a large Brown or Polar bear with a .22 - if the bear was chained down, you were allowed to reload several times, and you put the gun to the back of the bear's head.
In the wilderness? forget it.
Lots of large brown and polar bears have been killed with the .22 rimfire.
Most are killed with a single shot to the head. I have not documented polar bears killed with a .22, but there are many stories of Inuit doing so.
Quite a few big bears have been killed with .22 shots to the body, but it takes time to take effect. There is even a case of a black bear sow killed inside a city limits with a .177 air rifle shot to the body. It perforated a rib and the lungs.
The problem with using a .22 rimfire is the shot placement has to be precise. The target areas for a central nervous system shut down are fairly small. Bella Twin knew exactly where to aim. She took down a world record grizzly bear with a single shot .22 in 1953.
If facing a bear in defense, I would rather have a 12 gauge with slugs than a .22 handgun. But a .22 handgun you can shoot well is not to be despised. It has much better potential than a knife. The noise alone has worked for warning shots.
https://www.ammoland.com/2026/02/colorado-black-bear-killed-with-pellet-gun-in-2024/