Even though you cannot see it so well on the adult animals: Elephants have hair, especially on the head and the back. You can see the funny fuzzy hair on baby elephants particularly well.
Asian elephants generally have significantly more hair on the head and back than their African relatives. The very fact that they have hair at all, is surprising. After all, a thick fur is well known for keeping you warm, which seems rather impractical in the tropical heat. Their hair is, however, very fine. When a fresh gust of wind blows through the thin fuzz, it cools them down. Elephants even have hairs on their tusks. Just like cat whiskers, they serve to sense the environment.
Whereas there are around 500,000 - 700,000 elephants living in Africa, there are only just under 40,000 animals in Asia.