I also wish that I could get some ammo which is guaranteed to lose it's 'grunt' after a short distance. I'd hate to have a stray bullet cause some damage... and I hunt in some areas where you can't be sure that there is nobody else around, so extreme care is needed.
I use a lot of Winchester Subsonic hollow cartridges...and these are excellent. But they are still powerful and seem to have a long range.
I also have a few cartridges called Winchester Long Z. These have a full sized cartridge case...but a small load and a small projectile. I think they are designed to behave like the old-fashioned 'shorts', but they are longer to help make them fit a standard magazine and feed properly etc.
They say on my ammo packets generally that .22 bullets will travel up to 2 kilometres.... well over a mile. Frankly I think the chances of that are minimal in most cases involving a stray bullet, but I think that we have to consider that a bullet could go this far and get us into trouble.
I do not recommend a .22 rimfire as a gun to use on larger animals. But sometimes they are the safest and quietest option for hunting in certain places. This year I have shot 10 small pigs and a deer with a .22. The deer was head shot, and it dropped the quickest of any animal that I can remember. Three of the pigs I shot made it a short distance into the scrub, but I found two of them dead after a search. I admit that one is still lost, and I feel bad about it, but the scrub was very thorny and dense and I didn't have a lot of time. However I am fairly sure that the shot was a good one and it wouldn't have gone far before expiring. I've also had big animals run quite a bit when well shot with much more powerful guns. I used mostly Winchester subsonic for the pigs, and the deer was taken with a single round of Peters standard .22. Bullet placement is of prime importance when using a small caliber. And I would not normally use a .22 by choice for pigs and deer. It may even be illegal in some areas.
The rifle in the picture is an old Browning semi-auto fitted with a moderator. Beautiful gun, but as with any gun with a tube magazine you have to be heck of a careful to make sure the mag is actually empty when you are trying to unload it.
Where the use of a gun might be frowned upon, or downright stupid, I've used a bow and arrows and traps to harvest game. I heartily recommend the humane use of traps and snares...but I generally keep my operation very quiet so as not to upset anybody.