Apparently, the answer to this question is YES. It’s called “cold urticaria“. It’s a fairly new diagnosis, and researchers are not sure how many people have it.
If you have it, you’ll break out in hives after being exposed to cold air or water or even walking into an air-conditioned room. Sometimes, your hands will swell after holding a cold beverage or your lips will swell after eating cold foods.
Although not perfect, there is a test for this which is to hold a melting ice cube on the skin, remove it, and wait for a few minutes to see if hives begin.