Final answer:
Severely constricted blood flow to the skin in extreme cold can result in frostbite, which can cause permanent tissue damage and may lead to amputation in severe cases.
Step-by-step explanation:
If blood flow to the skin becomes severely constricted in extreme cold, the condition that can occur is frostbite. When body temperatures drop, the arterioles in the skin constrict to minimize heat loss, particularly in the extremities such as the fingers and toes. If the skin temperature drops too much, especially in environmental temperatures below freezing, the preservation of body core heat can result in the skin tissues actually freezing, leading to frostbite. This can cause permanent tissue damage and, in severe cases, may lead to gangrene and require amputation.