WASHINGTON—The Italian government last month quietly began allowing armed American drones to fly out of an air base in Italy for military operations against Islamic State in Libya and across North Africa—a breakthrough for Washington after more than a year of negotiations, U.S. officials said.

But Rome’s green light came with the kind of caveat that has crimped U.S. attempts to win allies’ help in fighting the extremist group: The Italians granted permission for the drones to be used only defensively, to protect U.S. special-operations...