If you are referring to the island in the Aegean Sea, its most usual English name, as far as I know, is just "Santorini" (no hyphen, no article, no "Island"). If the word "Island" was included then it would have a capital as it would be considered part of a proper name.
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TinaMrThank you, GPY! My main point in this question was whether or not I have to use definite article in this case.When the word "Island" is part of a proper name, the word is capitalised and there is no article. For example, "We visited Ellis Island".