I've never heard him say much on the matter. As he is a logical man, I'd say he likely does not, but he does "know" more than the average human. He once corrected Amy by claiming that some so-called "fairytale" creatures actually existed.
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tardis.wikia.com/wiki/Santa_Claus There was one episode where Santa actually showed up, but he turned out to be a dream. However, the "References" section on that wiki article contains one or two things that imply he believes in Santa, particularly the picture of himself, Santa and Einstein together.– IxrecCommented Dec 21, 2015 at 9:07
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2Then there's this from "The Doctor Dances" -- ROSE: Look at you, beaming away like you're Father Christmas. / DOCTOR: Who says I'm not, red bicycle when you were twelve?– RogerCommented Dec 21, 2015 at 14:27
2 Answers
It depends on the Incarnation.
In "A Christmas Carol" (2010 Christmas Special), the Eleventh Doctor says this:
The Doctor: Father Christmas. Santa Claus. Or, as I’ve always known him, Jeff.
This implies that he knows him on a "personal" level, but seems more like a throw away line.
As Roger says in the comments, the Ninth Doctor claims he MIGHT be Santa Claus himself:
Rose: Look at you, beaming away like you're Father Christmas.
The Doctor: Who says I'm not? Red bicycle when you were twelve?
And finally, the Twelfth Doctor realizes he is in a dream only because Santa Claus is in the base with them. At the end, when they're flying over London, they have this exchange:
Santa Claus: You want to take the reins, Doctor?
The Doctor: You're a dream construct, currently representing either my recovering or expiring mind.
Santa Claus: Yes, but do you want a go?
The Doctor: Yeah. All right.
So we might say some incarnations of the Doctor believe in him, while others don't.
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1Interestingly, none of those particular quotes necessarily contradict each other. Taking Nine at his implied word, then of course Twelve would know that the other guy isn't the real Santa, the Doctor himself is Santa, not this external entity. And that suggests that Eleven has revealed his own name to the viewers: Jeff. Doctor Jeff. But of course, the Doctor lies, and Nine used some fancy wording that doesn't even actually confirm anything about him being Santa.– 8bittreeCommented Jul 26, 2016 at 0:34
The doctor is kind of a fairytale himself. I don't think he started off believing in Santa because he was shocked when he saw Santa in the 2014 Christmas special when Santa had satsumas 😂