Canon is a fan-based idea that exists in a unique way within Doctor Who fandom. but it is more commonly thought as what a fan considers what forms part of the Doctor Who universe, or what "really happened". Despite this being a personal choice, it has been discussed and argued about in practically every Doctor Who-related forum or message board that has existed on the internet.
A literary canon of Doctor Who
In academic theory, "canon" refers to a body of work that an established body of literature that can drawn upon by future installments in the same broader tradition. In this sense, Doctor Who objectively has a canon, in the sense of later stories drawing upon the concepts and imagery of earlier works.
A good demonstration of this principle may be the mythos of Peter Cushing's Dr. Who; Dr. Who and the Daleks and Daleks' Invasion Earth 2150 A.D., due to being retellings of the TV stories The Daleks and The Dalek Invasion of Earth which brazenly contradicted the televised version of events, are often considered "extra-canonical" in the continuity sense of the term. However, these stories still "exist" and have not been ignored by even the BBC; a short story starring this Dr. Who notably appeared in the BBC Books short story anthology Short Trips and Side Steps featuring Dr Who, and critics have noted clear influences by the movies on the imagery of the Steven Moffat era of Doctor Who, from the vibrant colour of the Doctor's TARDIS to the fairy-tale atmosphere to even the details of Matt Smith's physical performance as the Doctor owing something to Cushing's.
As a narrative history, the fact that it exists is enough to consider it "part of the canon", in the sense that elements might make their way into future productions. The Dalek spacecraft of The Daleks were worked into CGI replacement shots on the DVD of The Dalek Invasion of Earth, and then further into television stories such as The Parting of the Ways. Again, this in an example of continuity within the show rather than as an established canon.
Acknowledgement by the BBC — or lack thereof
Unlike the principal rights-holders of other popular science-fiction and fantasy universes, such as Star Trek, Star Wars or Lord of the Rings universes, the British Broadcasting Corporation has never made a pronouncement about what is or is not canon for Doctor Who. Shortly after he brought Doctor Who back to television in 2005, Russell T Davies specified that canon "is a word which has never been used in the production office, not once, not ever". Even when specifying that the television show was going to contradict certain revelations about Time Lord history and biology from the Virgin New Adventures, Steven Moffat declared that those novels were "a separate" but "equally valid continuity".
In 1999, Big Finish Productions secured the licence to produced Doctor Who audio dramas. Stephen Cole was appointed as executive producer for the BBC to oversee Big Finish's content. In Doctor Who Magazine issue 275, Cole said of Big Finish:
As far as the BBC is concerned, these new stories are seen as part of the official Doctor Who canon. A great deal of responsibility comes with that status, and Worldwide did not assign this licence without careful thought.
A rare exception to this is in regards to video games: in August 2010, a BBC press release stated that, in Doctor Who: The Adventure Games, "Players will encounter new and original monsters, in stories which form part of the overall Doctor Who canon". Similarly, in July 2018, BBC Studios announced that, with games like Doctor Who Infinity, they would be "taking content from our major brands and delivering gaming experiences that actually form part of the canon. It's not led from TV series it comes to the game first." Another factor regarding the BBC's lack of an "official canon" for Doctor Who is that the BBC would simply not have the proper authority to declare one, as they do not, strictly speaking, own the Doctor Who universe; while they control the trademark "Doctor Who" and the copyright of a handful of key concepts and characters (such as the Doctor, the TARDIS or the Time Lords), a staggering amount of essential building blocks of the DWU are (or once were) in the control of individual rights-holders, including such elements K9, the Daleks or the War in Heaven mythos. As Paul Cornell noted
[The modern fandom notion of "canon"] works fine if you’re dealing with works by one author. It works not at all in any other frame of reference. Doctor Who was created by many people, over a long period of time, and they did not cooperate. There is no authorial authority, and […] no council of Bishops.
As close to executive non-canonisation as the BBC ever came was the effective "disowning" of the webcast Scream of the Shalka over the course of its production. 2003's Scream of the Shalka was to have been the continuation of Doctor Who, with Richard E Grant promoted as the "new" Ninth Doctor, and was written in that spirit; The BBC's first edition of Doctor Who: The Legend even has several pages which details the "Ninth Doctor". However, some time before release, plans for Russell T Davies's live-action revival of the series kicked into gear and prompted the BBC to cease all advertising of the "Shalka Doctor" soon after, due to the understanding that Russell T Davies intended to cast his own Ninth Doctor.
In the foreword fo The Nth Doctor, Jean-Marc Lofficier argued for the "canonicity" of even the unproduced 1990s script for a Doctor Who movie, although granting that the concept was "subjective". According to him, their occasional contradictions, albeit radical, to earlier stories, should not overshadow their clear intent to serve as continuations to the TV series, and, coupled with their having been approved by the BBC, this should allow them to stand as "canonical" to the same extent as other continuations like the Virgin New Adventures.
Argument about each 'Nth Doctor' script's degree of 'canonicity' will ultimately depend on each reader's own evaluation. (...) Since these scripts were not produced, their status as regards their 'canonicity' is highly subjective. However, one should bear in mind that these scripts wee fully licensed and approved by the BBC. The new elements that they proposed to introduce in the Doctor Who universe may often seem fairly radical, but (...) change, often radical change, has always been a respected tradition of Doctor Who. For these reasons I feel justified in treating the 'Nth Doctor' scripts as, at the very least, something closely related to main Doctor Who contnuity, not unlike the New Adventures.
Paul Magrs argued that a large issue when attempting to construct a definition of canon for Doctor Who is that it is never finished; between its many stories across practically every medium, Doctor Who has been in more or less constant production in one way or another since 1963. Some fans may want a complete narrative, but Doctor Who can never be complete; therefore "canon" is a non-starter.
At a 2008 San Diego Comic-Con panel, Steven Moffat remarked, "It is impossible for a show about a dimension-hopping time traveller to have a canon", laying the foundation for one way for "all stories to be true": rampant time travel and dimension-jumping combined to allow seemingly-contradictory stories to make up a single reality. Paul Cornell later wrote an essay on his blog in which he accused "canon" of being a reductive concept which primarily boils down to an excuse for fandom quarrels, highlighting how "‘non-canonical’ is a term of abuse in Who circles. A threat. It’s the worst thing someone can say about a televised Who story, that they regard it as not having ‘happened’." 
Echoing a similar sentiment, Nate Bumber released a short essay in which he compared the conception of canonicity in Doctor Who to the original context of terms such as "canonicity": the history of the Abrahamic religions. Bumber's fellow Faction Paradox writer Jacob Black pioneered systematic use of the more positive term "canon-welding" in online Doctor Who circles, treating "canon" not as quantifiable data, but as raw material to be "welded" into new patterns of continuity.
Ian Winterton suggested in 2021 that if there were a definition of "Canon", it should boil down to "whatever can be accessed via the BBC Licence Fee" — but that this did not devalue spin-offs, "tangential" as their connection to the BBC's Doctor Who might be. Winterton, however, admitted in the same interview that his definition did not seem to be foolproof, considering references to non-TV media on television such as Abslom Daak's cameo in Time Heist or the Eighth Doctor's "regeneration speech" in The Night of the Doctor which mentioned a few of his Big Finish companions. Big Finish themselves put out a joking tweet in March 2021, parodying the UK National Census with a question "Is it canon?" which was answered as "Other" rather than "Yes" or "No", and appended with a "helpful note" stating, "This question is going to cause an argument".
John Dorney drew a distinction between "canon" and "continuity", in that each audience member is free to choose which stories to accept in their personal continuity, whereas canon simply means "the recognised ‘official’ body of work. That ‘generally regarded as true’." "I don’t have a ‘personal canon’. I think the canon exists and it’s basically all the TV episodes, as they’re basically what everyone agrees on. I have a ‘personal continuity’, which includes the audios, the books, the DWM strips. No idea how they fit, but they’re in there."
In April 2021, Nicholas Briggs responded to a fan who about whether the Big Finish audio stories were canon, with Nicholas stating outright, albeit in a unserious way, that the stories were canon.
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