Can a body part fall because of frostbite?

fall off?
not really, but if left untreated eventually.

fail? as in not work anymore?
yes

frostbite will stop the flow of blood to a body part. it affects the smaller exterior parts first
(ears, fingers) and move on to larger parts (legs arms etc)
the colder you get the more your body will shut off blood to the extremities to protect the major organs ( heart,lungs,brain etc)

the body parts will turn black as frostbite progresses, if untreated quickly the body parts will die and freeze.