Nature has a news blurb about a find of mammoth bones that are only 5700 years old, in the Pribilof islands off Alaska. The pit on St. Paul Island contained the bones of animals which had fallen in and starved. Among these were a few bones of a smallish mammoth.
Mammoths became extinct in the Americas around 11500 years ago, which corresponds very approximately to both the arrival of humans and the last glacial minimum. But it seems that on the islands much more recent megafauna remains can be recovered-- see also sabre-toothed tigers on Hispaniola. The islands go through the same climate change but got colonized by humans later. Thus the trend suggests that people, not climate, were the change agent.
Mammoths were in Siberian islands even more recently, as recent as 3700 years ago.