We have the actual example of the Ice Man -- beaker culture neolithic discovered in the Alps. He was about 9000 years old.
So that can set some parameters:
Hunter or band traveling across and icefield that is in a pocket between multiple peaks. Dies in late fall. Exceptional snowfall period, so is buried under permanent snow cover which as the ice builds up puts him deeper.
Because it's a basin, the ice there doesn't move. When the ice builds up in thickness, there is a shear plane above it. Can't have glaciers feeding into the bowl either. So take a bowl that is surrounded by steep peaks that can avalanche into the bowl, but not form glaciers. Eventually the ice gets thick enough that it is above the bowl, and the excess flows off.
I think the Columbia Icefield on the border between Alberta and BC fits these criteria.
To me this would be a plausible event going back to the start of the last ice age. This would get you back to somewhere between 115,000 and 22,000 years ago.
Last Glacial Period -- Wikipedia For the longer period, you need to have some glaciation through the short inter-glacials. A mountain based shield would meet this requirement.
If you want a story line, "My father's father said that he knew a hunter who had crossed The Pass of the Winds." But in the hundred years since the cooling climate has created a permanent snowfield there.
Much of Antarctica is under water. Strip off the ice, and you have substantial inland seas. Antarctica without ice You could have various things trapped by glaciers closing off the openings. I don't know to what extent these levels freeze down. If there were hollows above the then present sea level you could trap land animals as described above. This could plausibly take you back to the beginning of the Antarctic ice cap something like 40 Megayear ago. I suspect that the timing of that is vague, so you might plausibly extend it back some. Dig into Antarctic paleo climate.