Frostbite while making ice cream?

The other day in chemistry we did the make ice cream lab for a fun practical application of freezing point depression. We had to kneaded the ice cream as we made it. I have poor circulation in my hands and we had nothing to protect our hands. After 3 minutes I had to put it down. Later I noticed my little finger was swollen. The next day it was kinda numb in a stingy way. Is frostbite possible in 3 minutes of holding ice in a bag at negative 5 degrees Celsius?