Octopus

Octopuses were a species of cephalopods (TV: Terror of the Autons) native to Earth among other planets.

On Earth
Professor Zaroff had a pet octopus. (TV: The Underwater Menace)

Julia Fetch admired octopuses and collected glass sculptures of them. (PROSE: The Drosten's Curse)

Elsewhere
On one planet, John was once grabbed by an octopus which restrained him with its tentacles. He was freed by the First Doctor when he tickled it with a feather. (COMIC: Shark Bait)

The Third Doctor compared the Nestenes to cephalopods, specifically octopuses. (TV: Terror of the Autons)

There were methane-breathing octopuses on Neptune in the 32nd century. (PROSE: The Eye of the Tyger)

Keraniums looked similar to octopuses. (AUDIO: "Incubation: Know")