On June 14, 2008, about 0845 eastern daylight time, a Cessna 350 (Lancair LC42-550FG), N166DS, registered to and operated by a private individual, experienced a shimmy following landing at the Elizabeth Field Airport (0B8), Fishers Island, New York. The personal flight was operated under the provisions of 14 Code of Federal Regulations (CFR) Part 91, with no flight plan filed. Visual meteorological conditions prevailed at the time of the accident. The Airline Transport Pilot and two passengers reported no injuries and the airplane sustained substantial damaged. The flight originated from the New Castle Airport (ILG), Wilmington, Delaware, at 0720. According to the pilot, the touchdown was normal and braking was applied. As brake pressure was increased, the airplane began to shimmy. Brake pressure was reduced slightly in an attempt to reduce the shimmy, with no effect. Full braking was then applied. The shimmy became more intense as the aircraft speed decreased to a stop. The pilot stated that taxi to the ramp was normal; however, postflight inspection revealed damage to the empennage, gear struts, and main wheel fairings. The accident was not reported to the National Transportation Safety Board until July 7, 2008. Examination of the airplane by a Federal Aviation Administration inspector and representatives from the Cessna Aircraft Company revealed that the fuselage composite structure was breached, just forward of the horizontal stabilizer. The composite core was exposed, and encompassed about 60 percent of the circumference of the fuselage. Tap testing revealed a larger area of composite damage surrounding the exposed core. The landing gear wheel pants and attaching parts were also damaged. Examination of the nosewheel found no anomalies in the operation of the nosewheel turning, dampening and centering. The left wheel pant lower surface, forward of the tire cutout, was scuffed. Both main tires were flat-spotted. The brakes were operational and no anomalies were observed on the brake pads/discs. The flaps were down and electrically inoperative. No anomalies were observed to the flaps and attaching brackets. Tire marks, 7 feet apart were, were observed on the asphalt runway 12 centerline. The marks started approximately 1,030 feet beyond the approach end of runway 12. Runway 12 was 2,328 feet long and 75 feet wide. At 1,500 feet, or 828 feet of runway remaining, heavier straight tire marks appeared and then became wavy. At 1,770 feet, the wavy pattern intensified to a diagonal skipping pattern and continued to the white threshold line, where the left side became wavy again and the right side straight, but with a skipping pattern. The tire marks dissipated 100 feet beyond the threshold line. Light tire scuffing marks were on the centerline and edge of the asphalt, 200 feet from the threshold line, consistent with a left 180-degree turn. Review of a Cessna 350 Pilot Operating Handbook, Section 5, Performance, revealed that the short field landing distance, with 12-degrees takeoff flaps, at standard temperature, with no wind or obstacles, was 1,950 feet at an approach speed of 88 knots indicated air speed. The reported wind at a nearby airport, about the time of the accident, was calm.