Leo G, if you assume the islanders were all dropped off on the island on the same day and given the instruction to leave if they have blue eyes, you would be correct. Most versions of the puzzle leave this ambiguous, but for the puzzle to make sense we have to assume that the islanders have no memory of exactly when they came to be on the island.
Even though everyone already knows everyone knows (etc) that at least one person has blue eyes, on the day after the announcement everyone knows everyone knows (etc) that no one left the island that day, therefore everyone knows everyone knows (etc) there must be at least two people with blue eyes, and this continues every day until everyone with blue eyes deduces that there must be one more blue-eyed person on the island than the number he counted and at that point all the blue-eyed people leave.
@Allan Crossman
You’re right. I was doing some sloppy thinking.