13 January 1923: birth of Jack Watling, who played Professor Travers in The Abominable Snowmen (Second Doctor, 1967), The Web of Fear (Second Doctor, 1968) and Downtime (unofficial, 1995).
ii) broadcast anniversaries
13 January 1968: broadcast of fourth episode of The Enemy of the World. The Doctor and friends escape Salamander's base, at the cost of Fariah's life. Salamander visits the underground base where he has imprisoned his scientists.
13 January 1973: broadcast of third episode of The Three Doctors.The Second and Third Doctors confront Omega, and discover that his physical body has been destroyed.
13 January 1979: broadcast of fourth episode of The Power of Kroll. The Doctor realises that the fifth segment is part of Kroll, and retrieves it, destroying the giant creature; most of the miners are killed and the Swampies survive.
13 January 1984: broadcast of fourth episode of Warriors of the Deep. The Doctor regretfully wipes out the reptiles with Hexachromite gas. "There should have been another way."
iii) date specified in canon
13 January 1213: On the 13th day of the 13th year of the 13th century, on the 13th moon of the 13th planet of the 13th galaxy, the Doctor and Rose encounter the Triskaidekaphobes (in the September 2006 Doctor Who Adventures strip, "Triskaidekaphobia" by the ever excellent Alan Barnes).