Gallifreyan cat

There were such things as Gallifreyan cats. (PROSE: Human Nature) They were the original cats, genetically engineered on the order of the old solar engineers by their biological engineer underlings and seeded throughout the universe. The cats who remained on Gallifrey were intelligent and capable of communicating with their masters, (PROSE: The Legacy of Gallifrey) and the Tenth Doctor once claimed to Donna Noble that this was also the case of the cats on Earth, a fact merely kept secret from their human "owners" by the cats, but she did not believe him. (COMIC: Autopia)

On Gallifrey, they remained a popular pet, (AUDIO: Nevermore) especially for Lord Presidents. (TV: The Mark of the Rani, PROSE: The Legacy of Gallifrey)

Also native to Gallifrey were the humanoid Killer Cats of Gin-Seng. (AUDIO: Erasure, PROSE: The Return of the King)

Biology
The original brief given to the biological engineer was to create "a Calculating Animal with a Tail" (an acronym of which was CAT) intelligent enough to point out to any budding mathematicians that it was impossible to compute the square root of Minus 3. (PROSE: The Legacy of Gallifrey) They had thirteen lives, and slept through all of them. (PROSE: Doctor Who and the Krikkitmen)

History
The Calculating Animals with Tails were devised as one of the tools in the Gallifreyan people's attempts to guide and preside over the technological advancement of other planets and species so that they could set themselves up as their Gods. The cats were designed as the kind of creature who would instantly and instinctively tell any budding mathematicians not to bother researching the square root of -3 due to this computation being impossible. They were created by the biological engineers on the order of the solar engineers. (PROSE: The Legacy of Gallifrey) The Fifth Doctor once told Tegan that he, in his original incarnation, had "introduced cats into Gallifrey's ecosystem" alongside Ruath, a friend and contemporary from his Academy days. (PROSE: Goth Opera)

At any rate, the sapient, yet often silent, cats were adopted as popular pets and symbols of intelligence on Gallifrey. Council Leader Tussan, the last leader of the Gallifreyan Council, owned such a cat, who sat in on Council meetings, by the time the engineers Rassilon and Omega came before the Council to ask for finances to create time travel using the trapped energy of a black hole. Intending to insult Rassilon, Tussan answered that he would let his cat decide. To Tussan's surprise and dismay, the cat, after yawning ostentatiously, answered that Gallifrey should finance Rassilon and Omega's experiments.

Tussan's cat was still by the Arcalian Council Leader's side when he made to punish Rassilon for his insolence after, at a much later point, the outraged engineer tore the Sash of Rassilon to put the Eye of Harmony beyond the reach of the Council. Weakened by his exposure to the Eye and aged before his time, Rassilon collapsed, dying, as he was being taken away by the Chancellery Guard — which was when Tussan's cat leapt down to hiss in Rassilon's ear, telling him not to give up. Moments later, to no one's greater surprise than his own, Rassilon regenerated, the first such event in the history of Gallifrey.

After Rassilon used the clout awarded to him by his miraculous resurrection to oust Tussan and the Gallifreyan Council and take control of the planet, Tussan's cat dropped Tussan and became Rassilon's trusted pet and advisor. This set a tradition of the Lord President of Gallifrey keeping a cat as an advisor, which was not broken until (PROSE: The Legacy of Gallifrey) Lord President Pandad IV's cat was eaten by a mouse that had been enlarged by the Rani in an experiment. (TV: The Mark of the Rani)

Much later, after the Fifth Doctor relinquished his Presidency, (TV: The Five Doctors) Flavia became Acting President with the blessing of Rassilon's shade. One of her first decisions was to renew the old tradition by having an intelligent cat given to her as a pet and advisor. She named the cat after the Doctor. (PROSE: The Legacy of Gallifrey)

Gramps was a cat living in a retirement home in Florida. He had a dignified manner and appearance, with fur in patterns of dark brown and black that seemed to move and blur impossibly in the shadows, streaks of silver-grey around his face, and no front right leg. He led a group of other cats in a fight against a younger cat — apparently Gramps' past self — in which the younger cat lost a leg. (PROSE: Gramps) All this bore a striking resemblance to the history of Grandfather Paradox, the semi-mythical founder of the Gallifreyan cult Faction Paradox. (PROSE: The Ancestor Cell, The Book of the War)

The Thirteenth Doctor later investigated a "talking cat" in Ontario. (TV: Praxeus)

Behind the scenes

 * Gallifreyan cats were further mentioned by a partially-in-character Jodie Whittaker in an online Q&A organised by the magazine WIRED in 2020, where, asked how many lives Gallifreyan cats would have (considering that Earth cats supposedly have nine while Earth people have just one, but Gallifreyan people have thirteen already). Whittaker claimed that Gallifreyan cats had infinite lives. This conflicts with the assertion in Doctor Who and the Krikkitmen that Gallifreyan cats have 12-shot regeneration cycles, just like their masters.
 * Notably, the talking cat in TV: Praxeus was hinted to be a Gallifreyan cat, which was indirectly confirmed to be the case on the did-you-know page for Praxeus on the BBC America website.