On October 17, 2020, about 1742 Pacific daylight time, a Piper PA-28-140 airplane, N4673R, was substantially damaged when it was involved in an accident near Pierce County Airport – Thun Field (PLU), Puyallup, Washington. The flight instructor, one student, and one passenger were not injured. The airplane was operated as a Title 14 Code of Federal Regulations Part 91 instructional flight. The student pilot reported that they conducted an uneventful preflight, runup, and takeoff. They proceeded to PLU to conduct a few touch-and-go landings. The first was an uneventful soft field landing and takeoff. Since the traffic pattern was busy, they flew a wide pattern and had a slightly faster than normal approach to land. After the second touch-and-go they were climbing with full power through about 500 ft when the engine sputtered momentarily. The instructor took over the flight controls and continued on a left crosswind when the engine power reduced to idle. The instructor turned the airplane back toward the airport and ensured the fuel selector was on, throttle and mixture were full forward, and both magnetos on. The student pilot pressed the starter to see if that would help, but the engine continued to idle and the airplane continued to descend. Unable to reach the airport, the instructor performed an off-airport landing. The airplane touched down momentarily and bounced back into ground effect. It impacted a few trees before coming to a stop in a parking lot. The airplane sustained substantial damage to the right wing and fuselage. A postaccident examination of the engine did not reveal any anomalies that would have precluded normal operation. The engine was rotated by hand; continuity was established throughout the drivetrain, and thumb compression was established. The carburetor bowl was removed, and fuel was located within the bowl. When the throttle was moved, fuel expelled from the accelerator pump. A small fuel tank was connected to the right-wing fuel line and the engine was prepared for an engine run. The engine started normally and exhibited a little roughness but smoothed out within a few seconds. The fuel and oil pressure gauges indicated normal pressure. The throttle was manipulated, and the engine responded accordingly with no anomalies noted. The temperature and dewpoint were plotted on the FAA Carburetor Icing Chart. Conditions fell on the line between serious icing at cruise power and serious icing at glide power. The flight instructor reported to the FAA inspector that he did not use carburetor heat once the engine started to sputter.