On October 14, 2002, about 1355 eastern daylight time, a Piper PA-28-181, N5675V, was substantially damaged during an overrun at Block Island State Airport (BID), Block Island, Rhode Island. The certificated private pilot and two passengers sustained minor injuries. Visual meteorological conditions prevailed for the flight that departed Nantucket Memorial Airport, Nantucket, Massachusetts; destined for Freeway Airport, Mitchellville, Maryland. A visual flight rules flight plan was filed for the personal flight conducted under 14 CFR Part 91. The pilot stated that the airplane was in cruise flight at 8,500 feet, about 27 nautical miles southeast of BID. About 1330, the pilot and passengers observed smoke in the cockpit, and notified the Boston Air Route Traffic Control Center (ARTCC). The flight was then vectored to BID. The pilot performed an emergency landing to runway 28, a 2,501-foot long asphalt runway. During the landing, the airplane touched down fast, about 800-900 feet beyond the approach end of the runway. The airplane subsequently overran the runway and struck a fence. The pilot stated that during the diversion, due to the smoke, he suffered a burning sensation in his eyes and nose, a headache, and dizziness. Additionally, the smoke restricted the pilot's forward visibility. Examination of the wreckage by a Federal Aviation administration inspector revealed substantial damage to the right wing. The left wing and horizontal stabilator were also damaged. Further examination revealed that the plastic housing and associated wiring on the battery box solenoid relay had melted. The reported winds at BID, at 1355, were from 360 degrees at 12 knots, gusting to 16 knots.