The Klamath tribe of Native Americans have oral stories and legends about the collapse of Mount Mazama and the formation of Crater Lake, which happened due to a massive volcanic eruption 6,000 and 8,000 years ago. Source. So oral histories, legends and myths regarding mammoth or other megafauna might possibly survive for an equally long period of time.

From NATHPO (National Association of Tribal Historical Preservation Officers):

To the Navajo, the mammoth figured prominently in their story of creation. Changing Woman (Asdzaa nádleehé) married the Sun and bore two sons, twins, and heroes to the Navajo people. They were known as "Monster Slayer" and "Child-Born-of-Water". The twins traveled to their father the Sun who gave them weapons of lighting bolts to fight the dreaded “monsters.”

Native Americans have a specific way of revealing their historical knowledge. Their oral stories are often embellished with interactions between historical events and supernatural beings. In 1762, the Shawnee told John Wright about the big stone skeletons found along the Ohio River. They said the bones belonged to an immense animal, the "father of all buffalo," which had been hunted by "great and strong men" of the distant past. But the Great Spirit destroyed the huge animals with lightning. The Delaware elders told Thomas Jefferson a similar story; only they claimed that the gigantic animals were driving away smaller game, like deer and bear. This angered their god, who blasted the great beasts with lightning. Only their petrified bones were left, although some thought that the huge animals escaped to the far north.

How much is later myths to explain the origin of mammoth bones and how much is actual oral history regarding mammoths is hard to determine now. In other words, the narrative may be recent enough for the fossils of the present to be interpreted, or old enough to explain how they got there. I'm sure someone more knowledgeable than me will be able to come up with a more comprenehsive answer though.

There exists a hypothesis that a mythical monster of Australian Aboriginals called bunyip may be a cultural memory of extinct giant marsupials such as the Diprotodon, Zygomaturus, Nototherium or Palorchestes.

I read a hypothesis that gold-horned deer from Slavic folk tales may be a memory of now extinct Irish elk, based on the golden color of its antlers, and being described as very large (in a ritual poem, described as carrying a man and a woman in its antlers). Sorry, could not find the source for the life of me.