It does not need to be hot, for one to get sunburn.
Sunburn is the result of over-exposure to the Sun's ultraviolet radiation.
Skiers faces and hands are exposed to the Sun when skiing, particularly at high altitudes.
The snow also reflects radiation, which other surfaces, such as grass and rock, would absorb.
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Yes, definitely. Since the sun reflects snow back up at you, and you are usually at a higher elevation when skiing, you might even burn more easily.