There are .20, .22 and .25 caliber pellet guns out there that are every bit as powerful as a 22lr... And in most places no licence, permit or what not is need, although I live in the backwards unAmerican state of IL where many high powered BB guns and Pellet guns are considered firearms and...
FYI you don't need a 'real' firearm to dispose of a tree rat... A BB or pellet gun will do the trick, mostly silent and generally legal to discharge even in a neighborhood, and little hassle to purchase in most areas if you are an adult...
Another option is a blowgun, you can pick one up at...