Some islands can sink, but they're much more temporary islands... every one of them does sink, or at least smack against a landmass for a very long time.
We had a little floating island of about 10 square feet drift across the lake once
Never sank, though... ran aground in a shallow part of the lake, and some silt built up around it. Now it's just a clump of dirt and grass and moss sitting on the ground... BUT IT IS STILL AN ISLAND! 
A... really small one... but really, this is just pedantic: we understand your point.
We had a little floating island of about 10 square feet drift across the lake once
Never sank, though... ran aground in a shallow part of the lake, and some silt built up around it. Now it's just a clump of dirt and grass and moss sitting on the ground... BUT IT IS STILL AN ISLAND! A... really small one... but really, this is just pedantic: we understand your point.
Please give me a home where cloud buffalo roam
Where the dear and the strangers can play
Where sometimes is heard a discouraging word
But the skies are not stormy all day
