Reindeer are almost mythical creatures. They are associated with Santa Claus and sleighs, with the idea of a Scandinavian icy white Christmas that is far more magical than the reality we normally experience in warmish, wettish Britain. But for me there's also something very special about reindeer because they are survivors from the Ice Age, clinging on when so many other magnificent large mammals died out at the end of the Pleistocene, through climate change or human hand or a bit of both. They are animals that were important to our ancestors, and animals that are still revered by the Siberian tribes who have a long history of hunting and herding them.

I first visited the icy north of Siberia five years ago while making a BBC documentary about ancient human migrations. We were filming with indigenous Siberians of the Evenki tribe, and staying in a remote reindeer-herders camp – living in tents that were kept warm with larch stoves while it was a bone-chilling -40C outside. (The stoves went out overnight and in the morning I would wake up to find my eyelashes stuck together with ice.)

There were reindeer all around us in the snowy, sparse larch forest. At night, they came in, walking cautiously around our tents, the thick fur behind their large hooves muffling their footsteps. One morning I wandered off into the forest to answer a call of nature. A single pure-white reindeer followed me. I wandered further and further with the reindeer following me a few paces behind. It felt as though I had made some kind of connection with this beautiful, ethereal creature. After I had done what I'd come for, I started to make my way back to camp, and wondered if the reindeer would follow me back. He didn't. Instead, he started tucking into the yellow snow I'd created. The mystical moment was shattered. He wanted nothing more than a few salts from my urine. Later I discovered that this apparently common behaviour was enshrined in a Siberian myth about the domestication of the first reindeer: a woman who went for a wee managed to catch and tame a reindeer who, like mine, had been after the yellow snow.

Long before humans domesticated reindeer, though, they hunted them, and reindeer-hunting was part of the reason our ancestors were able to colonise the polar north. We know that 30,000 years ago, modern humans – Homo sapiens like us – were living right up on the coast of the Arctic ocean. At one Arctic site, archaeologists have found ivory and horn spear-ends, stone tools and huge quantities of butchered animal bones, most of them reindeer. Those hardy hunters were surviving in extreme conditions: the average temperature in Siberia back then was even colder than it is today. We can only assume that those ancient Siberians made good use of reindeer fur in the same way that the modern Evenki do, to make clothes and boots. They would have been silly not to. I was given a pair of Evenki reindeer boots, and they are far warmer than the felt-lined expedition boots I was sent to Siberia with. The thermal properties of reindeer fur are legendary: it has a thick felty layer of underwool and long thick guard hairs that appear to be hollow, but on inspection under a microscope each hair turns out to have a core of vacuolated, keratinised cells.

Our European ancestors painted a great range of animals in their cave-cathedrals, but there's no doubt that they were particularly partial to reindeer. In some Aurignacian sites, 99% of the animal remains are from Rangifer tarandus, and this predilection for reindeer seems to distinguish our own ancestors from those other ancient denizens of Europe, the Neanderthals. As for what reindeer like to eat, you might think it was reindeer moss. But that's a misnomer – reindeer moss is in fact lichen. And while reindeer eat plenty of it in the winter months, digging through the snow with their broad feet to get at it, I expect that they are happy when the summer arrives and there are far more juicy and delicious plants to eat.

The end of the ice age saw a massive global extinction: many large mammals fell prey to changing climate and the effect of some rather formidable hunters sharing their landscape. There was a decline in the genetic diversity of reindeer after the peak of the last ice age 20,000 years ago – probably due to a warming climate as well as those palaeolithic hunters – but reindeer survived, and thankfully they don't look likely to become extinct any time soon.

I think our ancestors appreciated reindeer as beautiful creatures, as well as a source of food. The hunter-herders of Siberia still revere the wild reindeer, and give them sky burials on platforms in trees after hunting, killing and butchering them. One of the most extraordinary pieces of palaeolithic art is a 13,000-year-old ivory carving of two reindeer, found in Montastruc in France, but now in the British Museum. Some experts have interpreted them as swimming reindeer because their heads are thrown back and their antlers lie along their necks – but they have to be that way to fit into the tapering tusk. When those reindeer were carved, western Europe would have been a very different place, emerging from the ravages of the last ice age, and that small sculpture is a connection to a lost world. There's so much more to reindeer than red noses and sleigh bells.

The swimming reindeer carving features in the British Museum's forthcoming exhibition, Ice Age Art