Island
An island was a body of land completely surrounded by water.
On Earth
The United Kingdom was composed of the island of Britain, (TV: The Stones of Blood) better known as Great Britain, (TV: The Empty Child, Rise of the Cybermen) and Ireland, (TV: Tooth and Claw) forming part of the British Isles (TV: Boom Town) or British Islands, (TV: Fury from the Deep) which included the Shetland Islands. (TV: Army of Ghosts) There were also islands off the coast of Wales. (TV: They Keep Killing Suzie)
St John's Monastery was built on a "tiny island" on Earth in the 13th century, standing as late as the 22nd century. (TV: The Rebel Flesh/The Almost People)
Manhattan was an island part of New York City in the United States of America. (TV: Evolution of the Daleks)
Japan was composed of islands. (TV: Last of the Time Lords)
Azores was a group of islands which included Santa Maria. (TV: The Chase)
The South Island was part of New Zealand. (TV: The Tenth Planet)
Anuta Island was located in the South Pacific. (TV: Resolution)
Irving Braxiatel created a floating island which he dubbed Laputa, stationing it near Venice, Italy in 1609. The Armageddon Convention was held there. (PROSE: The Empire of Glass)
The Fourth Doctor and Leela visited Easter Island in 1872. (PROSE: Eye of Heaven) River Song said the Eleventh Doctor was worshipped there when they visited it. (TV: The Impossible Astronaut)
There was a Silurian colony beneath the Galápagos Islands, which revived during Charles Darwin's visit in 1835. (AUDIO: Bloodtide)
Jack Harkness visited Ellis Island at least twice. The first time was in 1892, where he experienced his second death and realised he was immortal. (TV: Utopia) The second was in 1927, where he met Angelo Colasanto. (TV: Immortal Sins)
A lone Dalek fell to Earth on the Ascension Islands. (TV: Dalek)
After the First Doctor and Vicki Pallister accidentally destroyed his drawings for New Rome, Caesar Nero raged "I'll stick you both in the arena, on an island with water all round, and in the water there will be alligators and the water level will be raised and the alligators will get you!" (TV: The Romans)
Elsewhere
An island was found on Marinus. (TV: The Keys of Marinus)
Fire Island was located on New Earth. (TV: Gridlock)