Pinocchio

You may be looking for the Faction Paradox member.

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)