James T. Kirk

In a parallel universe, James T. Kirk was the captain the USS Enterprise in the 23rd century.

Biography
On one occasion, the Eleventh Doctor recalled memories of his fourth incarnation meeting up with Kirk and the command crew of the USS Enterprise to combat a CyberNomad invasion of Aprilia III. After Kirk had informed the Doctor that his communicator had a gold cover, he distracted the Cybermen until the Fourth Doctor used the gold to disable them. While the Doctor took his leave, Kirk checked on the staff, blissfully unaware that Cyberman coding still existed in the facility's systems. Following the Eleventh Doctor and the Enterprise-D defeating the cyber-web, the Doctor suspected that his encounter with Kirk had been erased from the timelines. (COMIC: Assimilation²)

Kirk was seemingly among the visitors to the Dalek Dome on Earth, alongside Mr Spock in 2323. (COMIC: Liberation of the Daleks)

Other references
A fictional character named Captain Kirk appeared in the science fiction television series Star Trek in the Doctor's universe. (PROSE: The Gallifrey Chronicles)

Spock's Brain was an episode of Star Trek which both Izzy Sinclair and Destrii had seen. Destrii even went so far as to call it her "favourite." She cited "the bit where Sulu has to do the captain's log" as a particularly memorable moment and even quoted back a portion of the log, in unison with Izzy: "Captain Kirk's hunch that Spock's brain is on this planet appears to be correct!" (COMIC: Ophidius)

When Clyde Langer and Luke Smith were on Kudlak's ship Luke asked Clyde if he could use his mobile phone to which Clyde sarcastically asked if he had a phone number for Captain Kirk. Luke was unfamiliar with the character. (TV: Warriors of Kudlak)

Donna Noble briefly adopted the alias of "Captain Kirk" to the Tenth Doctor's "Dr McCoy" when dealing with a group of soldiers. (AUDIO: Pest Control)

When DI Patricia Menzies entered the Sixth Doctor's TARDIS and saw its console room in 2010, she commented that she had expected the TARDIS interior to resemble "Captain Kirk's ship". The Doctor responded, "Detective Inspector, may I tell you something and I can't stress this enough: I am not Captain Kirk." (AUDIO: The Crimes of Thomas Brewster)

In 2015, Cory said he became an astronaut to be like Captain Kirk or Han Solo. (COMIC: The Hyperion Empire)

Earth-33⅓
Jim Kirk also existed in Earth-33⅓. When Star Trek's Beverly Crusher won the Favourite Doctor Award at the Doctor Who Magazine Annual Poll, Spock told Kirk that it was "'acting', Jim[,] but not as we know it". (COMIC: Doctor Who? 176)

Behind the scenes
Assimilation² depicts Kirk with the likeness of actor, who originally portrayed him in Star Trek.

The mentions of Captain Kirk in The Gallifrey Chronicles, Ophidius and The Crimes of Thomas Brewster treat him as a fictional character (as do, implicitly, other references to Star Trek and Star Trek: The Next Generation in Doctor Who stories). Assimilation², of course, treats him as a "real" individual. It also shows that he exists only in another universe. It is thus possible that he is a character in the Doctor Who universe but a real man in the Star Trek universe.

Captain Kirk was briefly portrayed by Fifth Doctor actor Peter Davison in the closing sketch of an episode of The Harry Hill Show.

J. K. Woodward, who worked on the Star Trek crossover Assimilation², released a piece depicting Captain Kirk standing with the Eleventh Doctor and Amy Pond, the latter wearing a 23rd century Starfleet uniform and holding a tricorder.