It turns out that a few mammoths made it through the Ice Age. Wrangel Island is an uninhabited land mass off the northern coast of faster eastern Siberia. It's about the size of Delaware and according to fossils found there, until about 4,000 years ago, it support the world's last mammoth population.
For 6,000 years, 500-1,00 mammoths were alive on the island while their mainland cousins disappeared.This was round 1650 BCE, which is far more recent than the Ice Age. By this time, the Egyptian pharaohs had started the second half of their 3,000 year reign and the Great Pyramids of Giza had already been built.
It is hypothesized that these beasts may have died out due to inbreeding and a lack of genetic diversity (as the island was isolated and small). Archaelogical evidence suggests that humans reached the island around the same time the mammoths disappeared.
While this is a suspicious coincidence, no artifacts have been found in the area that suggests that the mammoths were have been hunted. It is more likely, that it was climate change finally found these last mammoths and killed them off.