Peter Capaldi (in-universe)

Peter Capaldi (PROSE: A Letter from the Doctor) was an actor who portrayed the Doctor in the BBC series, Doctor Who. (TV: In the Forest of the Night, PROSE: A Letter from the Doctor, WC: The Zygon Isolation) He thought that the Doctor was fictional, but was ultimately uncertain. (PROSE: A Letter from the Doctor)

An advertisement for Doctor Who with Capaldi was applied to the side of a bus in November 2014. (TV: In the Forest of the Night, PROSE: "Assessing the Risk")

On 26 April 2016, Peter Capaldi wrote a letter for and requested by Doctor Who Magazine, which they printed in their five-hundredth issue. Capaldi was also asked to write a letter for another issue, but he couldn't remember which one. (PROSE: A Letter from the Doctor)

Peter Capaldi appeared to play the Doctor in at least the ninth series, specifically in the episodes New Series Prologue, The Magician's Apprentice, The Witch's Familiar, Under the Lake, Before the Flood, The Girl Who Died, The Woman Who Lived, The Zygon Invasion, The Zygon Inversion, Sleep No More, Face the Raven, and Heaven Sent. (WC: The Zygon Isolation)

Alternate universes
An actor called Peter Capaldi also existed in a parallel universe where the adventures of the Doctor were solely fictional. The Eleventh Doctor suggested he play the Doctor. (COMIC: The Girl Who Loved Doctor Who)