PA-28 Cherokee Flight Instructor reported partial loss of normal braking system on landing. Instructor shut down the engine; pushed the airplane off active runway; and safely brought it to parking.

Narrative

Aircraft brakes were functioning normally during initial taxi. Instructor and student were conducting traffic pattern training with multiple touch-and-go landings. Initial departure was from Runway XXR. ATC later assigned Runway XXL for continued touch-and-go operations.Two touch-and-go landings and one full-stop landing were completed. During the final full-stop landing; the aircraft experienced partial brake failure. Braking effectiveness was reduced; and the aircraft was unable to stop at the assigned taxiway exit. The aircraft passed the intended exit and came to a stop slightly beyond it; positioned perpendicular to the runway.The instructor immediately informed ATC of the brake issue. ATC was advised that the aircraft was unable to taxi normally. Permission was requested to shut down the engine so the aircraft could be pushed back toward the taxiway exit. ATC approved the request.After the aircraft was repositioned; the engine was restarted; and the aircraft exited the runway via the closest taxiway. ATC was informed that the aircraft was clear of the runway. No injuries occurred; and no other aircraft were affected; aircraft safely brought back to parking