During an instructional flight, the student pilot and flight instructor remained in the local area to practice maneuvers. On return to the airport, during the first full stop landing, the student pilot landed in a flat attitude, but left of centerline. During the landing roll, the student pilot veered left onto an intersecting runway and the flight instructor applied the hand brake. The airplane overran the width of the intersecting runway into rough terrain, the nose landing gear collapsed, and the airplane nosed over resulting in substantial damage to the firewall and right wing lift strut. Both occupants reported no mechanical malfunctions or failures with the airplane that would have precluded normal operation; however, the student pilot stated that a gust of wind caused the airplane to veer to the left. The flight instructor's statement and weather reports indicate that there were no wind gusts at the time of the accident.