The titular character of the book was inspired by Alice Liddell, Lewis Carroll's neighbour in Oxford, (PROSE: The Wildthyme Effect) and according to one account, Wonderland indeed existed beneath the Earth's crust, inhabited by many of the characters described in the book. (AUDIO: Iris Wildthyme and the Land of Wonder)
The Third Doctor thought to himself, "As Alice once said 'This gets curiouser and curiouser,'" while being escorted by guards into the complex where the Gemini Plan was being carried out. (COMIC: Gemini Plan)
The Fourth Doctor misquoted Alice in Wonderland when coming to after being scanned in the Kraal Disorientation Centre, combining the Dormouse's story about three sisters living at the bottom of a treacle-well with Three Sisters by Anton Chekhov. (TV: The Android Invasion)
Peri Brown had been reading an omnibus edition of Alice in Wonderland aboard the Doctor's TARDIS before accidentally activating the Doctor's personal reality warp. The device transported her to a world largely based on the book, with herself cast as Alice. (COMIC: Salad Daze)
When the Seventh Doctor told Ace that he didn't know why the Cloister Bell was ringing, she replied, "You don't know much, and that's a fact." The Doctor recognised the quote from the Duchess in Alice's Adventures in Wonderland. (PROSE: Timewyrm: Genesys) On a later occasion when she was disorientated, Ace compared her situation to that of Alice. (PROSE: Lucifer Rising)
When the Eighth Doctor was infected by Zagreus, the Doctor's TARDIS gave him hints and aid through a copy of The Alice Compendium from the Doctor's library. The TARDIS also manifested a variety of environments and avatars to help him, among which was the story's forest and Cheshire Cat, who confronted the Doctor with a version of the story's "drink me" puzzle. An evil avatar of the TARDIS, manifested by Zagreus, later summoned a Jabberwock from the book to attack the Doctor. (AUDIO: Zagreus)
Behind the scenes
The actual title of Carroll's book is Alice's Adventures in Wonderland, though the shorter Alice in Wonderland is widely used.
Simon Russell Beale, Ken Dodd, Elizabeth Spriggs, Jason Flemyng, Joanna Lumley, Sheila Hancock, Murray Melvin, Nigel Plaskitt, Hugh Lloyd, Christopher Ryan, and Mary Healey appeared in the 1999 made-for-TV adaptation.
Matt Lucas, Stephen Fry, Barbara Windsor, Michael Gough, Tim Piggott-Smith, and Michael Sheen appeared in Disney and Tim Burton's 2010 live-action adaptation. Lucas, Fry, Windsor, and Sheen reprised their roles in the 2016 sequel.
Frank Welker also provided the vocal effects of the Bandersnatch in the latter film.