Pinocchio was a fictional character made of wood that wanted to "fit in".
Susan Foreman compared herself to him, citing her want to be a "real live girl" as the reason behind her choice to attend Coal Hill School while visiting Earth 1963. (PROSE: Time and Relative)
Bernice Summerfield, who was being manipulated by the Seventh Doctor, said that she was tired of having her strings pulled like Pinocchio, whom she knew from a Disney film. (PROSE: Birthright)
The Eighth Doctor compared moving wooden puppets created using Cyber-Technology to Pinocchio. (AUDIO: The Silver Turk)
When trapped aboard a submarine with a murderous Ice Warrior, Clara Oswald sarcastically remarked that keeping her spirits up by singing as Professor Grisenko suggested would only help "if this was Pinocchio". (TV: Cold War)