Reindeer

Reindeer were a mammalian species of the deer family, often associated with Santa Claus.

Encounters with reindeer
The Doctor, their companions and acquaintances occasionally encountered reindeer — and, indeed, Santa Claus himself. Once, Santa sent a sleigh pulled by reindeer to the First Doctor, John and Gillian to take them to his house. (COMIC: A Christmas Story)

On another occasion, the Third Doctor and Sarah Jane Smith met St Nick on a New York estate — an encounter which was witnessed by Clement C. Moore. (PROSE: A Visit from Saint Nicholas)

In a dream, three reindeer, Rudolph, Donner and Blitzen, flew around Clara Oswald's home after Santa's sleigh crashed on the roof. Santa later rode into a polar base on Rudolph to stop the dream crab attack. Santa claimed to Shona McCullough that his flying reindeer were a "scientific impossibility", and as a result, he fed them "magic carrots" so they could fly. (TV: Last Christmas)

At Hyperville — Europe's largest shopping mall in 2013 — the Tenth Doctor encountered the Snow Queen, who, like Santa, chose to travel in a sleigh pulled by reindeer. (PROSE: Autonomy)

According to one account, Santa's reindeer differed physically from normal reindeer, apparently being "a shade greener" in colour, and possessing at least two more legs. (COMIC: Frosty the Snowdemon)

Of course, not all encounters with reindeer were associated with Santa. The Eighth Doctor and Izzy Sinclair once met some Noxmycota-crazed reindeer when they visited Sweden in 1070. (PROSE: Illumination)

Cultural references to reindeer
Reindeer were so associated with Santa Claus that replicas were often made of them for Christmas decorations. The Tenth Doctor and Donna once landed at a "Santa's grotto" in a garden centre which contained large reindeer figures. (COMIC: Frosty the Snowdemon) Artists often included reindeer in their Christmas illustrations — just as Jasmine Deering did in her advertising campaign for Blackstone soft drinks. (PROSE: Christmas in Toronto)

Information from invalid sources
A Christmas stocking was decorated with an illustration of a reindeer. (NOTVALID: 2016 BBC Christmas ident)

Other matters

 * A reindeer was a "special guest" in several short, invalid promotional videos produced for BBC One in 2009 for broadcast at Christmas of that year. The spots feature David Tennant, in costume and in character as the Tenth Doctor, cavorting in a snow-covered forest with the TARDIS and a reindeer. In the longest video of the promotion, the Doctor hooks several reindeer up to the TARDIS and they take flight (suggesting they are flying reindeer like those used by Santa Claus). Two short "sting" promos of about ten seconds each show a single reindeer wandering outside the TARDIS.