Wrap the poor appendage in a damp cloth or paper towel, then place it inside a plastic bag of the ziploc variety. Squeeze out the excess air and seal the bag, then place the bag in cold water (ice water, if available.) If no plastic bag is available, you can wrap it more thoroughly in cloth and then place it on ice, but this is more likely to cause tissue damage (frostbite.)

Supposedly, if stored in this manner, a finger will remain salvagable for approximately 18 hours. Without cooling, it would have only been good for around 5.