I saw a report somewhere that a pumice "island" had some skeletons on it, indicating that it had lasted a while. I wonder if such pumice islands are the explanation for some of the mysterious islands reported by sober, reliable ship captains, islands that have never been found again. If the same undersea volcano erupted again several years later, that could explain the reappearance of some of those vanishing islands.
Quite apart from pumice islands, there are, sadly, man-made Garbage Islands comprising plastics and other human detritus that has coagulated together due to natural currents in the oceans. When you consider the relatively short time we have been using plastics, it becomes worrying for our future. Perhaps the word "raft" should be used rather than "island" as islands generally do not float
Whether islands float or not depends on what definition you use of the word "float". It has many definitions.
If you define "float" as "stay on the surface, be buoyant, be buoyed up" (as opposed to sink), then islands float, don't they? After all, they don't sink, do they?
On the other hand, if you define "float" as "move or hover slowly and lightly in a liquid or the air; drift", then I agree. Islands generally don't drift away in the water.
As for man-made islands, the Aztecs famously built a huge island in the midst of Lake Texcoco, and on this island they built Tenochtitlan, their capital city. But they also built many floating gardens called chinampas in the shallow lake beds of the Valley of Mexico, not just in Lake Texcoco but also in the nearby Lakes Chalco and Xochimilco. What would the Americas have looked like if - as nearly happened - Cortez was captured during La Noche Triste, and the Spanish conquistadors faltered?
Some islands are countries and some countries are islands, but no man is an island, although you do have the Isle of Man but not Isle of Woman (though Lesbos is a strong candidate), but then Isla Mujeres off the coast of Mexico also applies, as it translates as "the Isle of Women".
These are some thoughts that prevented me from pursuing my PhD.
I don't think its the definition of 'float' that's causing confusion here, but the definition of 'island', which is typically crust exposed above sea level. Unless you are referring to them not sinking into the mantle, but that's the same for all crust.