The force said marine officers on patrol found a 54-year-old man who was yelling for help in the sea on Wednesday night.
A rescue operation was launched with the help of the Government Flying Service and a man in his 60s was found unconscious in the water. He was later certified dead in hospital.
Police said their initial investigation suggested the two men were travelling from the mainland.
The survivor was arrested on suspicion of endangering the safety of others at sea and illegal entry.