Can a 22 caliber 5.5mm (pellet) airgun kill a pigeon with one shot?
Absolutely. If the shot is well placed (especially in the head) and not made from an inappropriately long distance it will easily kill a pigeon with one shot.
I am familiar with examples of people taking down white tailed deer with a single shot from a .22 air rifle, though these were undoubtedly illegal (due to local laws) and inhumane (due to the high likelihood of NOT killing the deer with one shot).
But a .22 air rifle is more than adequate for taking all sorts of small game like squirrels, some birds, raccoons, etc. Your local laws may allow for hunting with air rifles and may also list which prey are allowed to be taken with which caliber of pellet.