On the ground, residents fear the virus has spread beyond the handful of cases that may be officially recorded. In East Hampton, where cases more than doubled over the past week, to 83, Kamal Jackson, a volunteer firefighter, said calls from dispatch of “FC,” fever cough — now often code for potential coronavirus — have come over his transponder nearly every day for the last three weeks.

“It was imported from somewhere else,” said Mr. Jackson, 40, who also works for a pool maintenance company and does construction work. “I don’t fault them. They own property; they have a right to be here. But it was definitely brought out here from them.

Leaders from across Suffolk County, on the eastern half of Long Island, sent a letter to Gov. Andrew M. Cuomo on March 27, asking for official restrictions on travel to the area. The governor said on Sunday that he had not “heard any local officials raise concerns” about the need for more travel restrictions and had not yet considered introducing them.

The Shinnecock Indian Nation, a tribe whose leaders signed the letter to the governor and whose 980-acre reservation is at the westernmost edge of the Hamptons, is trying its own tactic: On the edge of Sunrise Highway, the tribe’s large electronic billboards scroll President Trump’s recent advisory that residents of New York, New Jersey and Connecticut “refrain from nonessential domestic travel for 14 days.”

“It feels like we have been invaded and they are taking advantage of our enclave,” said Bryan Polite, the chairman of the tribe. “Those people were selfish, and we have to deal with the ramifications of their actions.”

Donald G. McNeil Jr. and Azi Paybarah contributed reporting.