Trenchcoats were coats made for protection from weather, famously worn on wars.
Worn by the Doctor
The Fourth Doctor wore a single-breasted trenchcoat of plum purple velvet when he and Romana II foil Tryst's conversion of Mandrels into Vraxoin. (TV: Nightmare of Eden) He also wore a burgundy trenchcoat, (TV: The Leisure Hive) up to his death. (TV: Logopolis)
The Seventh Doctor wore a black leather trenchcoat on Nazi Germany. (PROSE: Timewyrm: Exodus)
The War Doctor wore double-breasted leather trenchcoats on black (COMIC: The Whole Thing's Bananas) or brown. (COMIC: The Clockwise War, The Lost Dimension, Four Doctors; TV: The Name of the Doctor, The Day of the Doctor)
On World War 1, the Ninth Doctor wore a British Army trenchcoat. (COMIC: The Forgotten)
The Thirteenth Doctor wore a lilac-blue trenchcoat with a navy lining. (TV: The Woman Who Fell to Earth, onwards)
Worn by Companions
Sarah Jane Smith occasionally wore trenchcoats. (TV: Eye of the Gorgon, The Last Sontaran, The Mark of the Berserker, etc.)
Amy Pond wore a dark teal trenchcoat during her encounter with Vincent van Gogh. (TV: Vincent and the Doctor)
Donna Noble wore a single-breasted trenchcoat of tan leather. (TV: Turn Left, The Stolen Earth/Journey's End)
Worn by the Master
The Spy Master wore a double-breasted trenchcoat of black leather when posing as a Nazi in Paris. (TV: Spyfall)
Worn by Others
Detective Duggan wore a beige trenchcoat when he helped the Fourth Doctor and Romana II. (TV: City of Death)
Sally Sparrow wore a dark brown trenchcoat. (TV: Blink)
The Second Rani wore a black trenchcoat when posing as Professor Baxton. (AUDIO: The Rani Elite, Planet of the Rani)