No - there could be a world covereed in land, with "islands" of water (aka lakes) alll over the place. As long as those lakes didn't connect in such a way as to cut off any part of land from any other part of land, you could have plenty of water with no islands. 
I suppose you are right there, but then suppose that there is a planet with a thin (2mm wide) strip of land running around the circumference of the planet at the equator, and the rest is all water, the water bodies would be lakes, right?

(not sure of the exact definition of "island," so correct me if I am wrong)
In other words, if an island both encloses water and is enclosed by water in some way, it is no longer an island.
