Well, actuall, Carol C., mink farmers kill and skin the minks and then feed the carcasses to the other minks. Minks are very nasty critters.
A good trapper will make a set where the animal will die, usually by drowning, or use the "instant kill" traps that are now available (made by Conibear, I believe). Professionals learned that a pelt damaged because of the attempts by the animal to escape is worth less than an undamaged one.