Not sure of the resort but no there is a tennis court on the North End of the Island. Perhaps Journeys End ??? I do not play but some folks I know have played up there. Sorry I could not have been of more helpbut I;m sure someone here can give you the name or the resort.
There are tennis courts (really nice, too), at the Essene Way, as well as the one at Journey's End. Maybe if you order one of Pauly's New York pizzas at the Essene Way, he'll let you play a few sets. Good Luck.