Desert Island Dick from The Topper was just a whole comic strip about this trope: one guy lives on a deserted island by himself and never seems to be able to get off or starve. He is friends with the wildlife though. This trope has also been occasionally used in The Topper's more well known stablemates The Beano and The Dandy.
MAD's Don Martin often used this trope in his comics. As does Frank Jacobs.
It also wasn't always deserted; rumrunners used it as a secret cache before the British navy cracked down on them.
However, in the third film one of these islands (sans the palm tree) is used as a quick meeting place for the heroes and villains to negotiate.
At the very end of Superman Returns, Lex Luthor and Kitty are shown to be stranded on a Far Side Island because they ran out of fuel for the helicopter.
Rincewind from the Discworld series enjoyed his time on one of these (between Eric and Interesting Times). It was boring and the Luggage kept itself entertained by fighting sharks.
The narrator in Terry Pratchett's Nation points out that in the alternative universe world, a type of palm tree exists that poisons all other palms on its island, "making all those cartoons botanically accurate".
John Sladek once wrote a short story lampooning cartoon stereotypes, and of course saved this one for last.
A Series of Unfortunate Events: Each book ends with the usual author/illustrator info, including a humorous self-portrait of Brett Helquist having something to do with the plot of the book. In The End, which takes place on a faraway island, Helquist draws a stereotypical portrayal of this trope.
Live Action TV
A sketch on The Muppet Show had Cloris Leachman stranded on an island with Sweetums. They are later joined by Doglion.
In a rare example where neither occupant seem to care much, Sesame Street's Count von Count and Harry Belafonte are stranded on a tiny island. But as long as The Count has a plethora of coconuts, he's good to go, and Harry seems content to sing about him.
New Zealand kids' show What Now?parodied this by having the cast marooned on a traffic island.
Private Eye did one once complying with gender, race, religion and disability discrimination laws, the comic in question featuring a woman in a burhka and a wheelchair on a desert island reading a message in a bottle rendered entirely in As Long As It Looks Foreign script.
Regularly featured in New Yorker cartoons, such as one that has the palm tree replaced by a wind turbine and one castaway saying to another, "I miss the palm tree too, but at least we can have a refrigerator." Lampshaded in this one.
Gahan Wilson did a bunch of tragic 'stock cartoons' for the National Lampoon, including a little desert island with a bleached skeleton on it.
The Onion in Our Dumb World had a wonderful section lampshading this, referring to the islands of Kiribati as "An island, a palm tree, and one guy" going through all the clichés of these types of cartoons.
There's a tiny island in the background of one section of Keelhaul Key in Paper Mario: The Thousand-Year Door. There's a palm tree there, and if you hit it with a hammer, a Coconut falls out for you to collect and exchange for the Chuckola Cola Flavio has on him, if that's where you are in the story. There's a pipe linking it to the main island for "easy" possible access.
Tales of Monkey Island has a few of these, especially in Episode 2. When Guybrush encounters one of these for the first time, he lampshades this with his remark "I didn't know tiny islands like this really existed." He also comments that he's glad he's not stranded there (he got there by sailing to it after choosing the destination from the ship's map) or else he'd have to worry about having something to read.
In zOMG!, Sandblast and his Elite Mooks can only be battled by taking a canoe from Gold Beach to one of these.
A viewer of Homestar Runner once asked Strong Bad what it would be like if he and Homestar were stranded on such an island. Strong Bad first mentioned the usual tropes for such a situation, but figured the reality would actually be more dull and unpleasant. In another episode, we see a flashback of him being saved by The Paper from a sinking version of one.
A Chinese animation called See Through revolves around two fighter pilots from opposing armies becoming friends and forming an Ambiguously Gay relationship on one such island after downing each other in the middle of combat.
Used in the short Sluggy Freelance side story where Bun-bun gets sidetracked trying to get to Tijuana to set up a black market Viagra ring. Stoner Gilligan provides Bun-bun with company... for a short time.
In Wapsi Square, Monica finds a small tropical island that more or less fits this description, to which she and the golem girls can poit for a bit of peace and quiet. Except that the last time Bud went there she found something else and the island got blown to bits.
One of these showed up on American Dad!. Originally it was a quite spacious desert island with many luxuries installed, until a tidal wave submerged something like 95% of the island underwater. Stan and Roger survive by eating mostly seagulls.
One of these is seen regularly on Spongebob Squarepants as an establishing shot for the city Bikini Bottom. The cast even went there in the episode "Pressure".
Showed up in an episode of Jimmy Two-Shoes. The only inhabitant was a weavil Lucius had imprisoned there.
Taz-Mania: Taz and Wendal are stranded on one in "Taz-Manian Theatre".
Rockall is perhaps the best real life example, although it is a single large rock jutting out of the North Atlantic, 162 miles away from the nearest landmass and sporting no palm trees. Most useful as a means to claim exclusive economic rights in the surrounding ocean, Rockall is normally uninhabited, but claimed by the UK, Ireland, Iceland and Denmark. Longest duration anyone has spent on the island is 42 days.
Of course, many islands are extremely small, but most this size are within swimming distance of larger landmasses.
Fort Jefferson in the Dry Tortugas (part of Florida) is practically built on a Real Life version. It's closer to 100 meters wide rather than just one, but the public non-fort area which allows overnight camping would fit the bill. ETA: It's about fifty miles west of Key West, making it one of the most remote spots in the continental USA.
There is a well known research site leased by Sydney University near the Great Barrier Reef that doesn't quite fit this trope perfectly as it's 5.5 km across at its widest point, but it's named One Tree Island.
A Cay is the term for a small, flat island that forms around coral reefs, when sediments pile up in one spot on the reef.
Can be an exaggerated Truth in Television for many of the countless coral atolls scattered around the globe. Atolls form from the remains of old undersea volcanoes which result in a ring shaped series of coral islands around where the mountain previously emerged from the sea. A single Atoll can be comprised of many small islands, some remarkably similar to the trope in its most literal sense, but even where the islands are large they are still small by any land based means of comparison. For example in World War 2 thousands of soldiers would fight over strategic coral islands the side of a good sized city park.