The certified flight instructor (CFI) stated he arrived at the airport to conduct a training flight with a new student pilot. A preflight inspection was conducted and the only anomalies noted was low pressure on the nose wheel. The tire pressure was increased and CFI and student pilot entered the airplane. The engine was started, the airplane was taxied to runway 24, and the required checklist items were completed. They departed the airport and flew north of the airport and completed some air work before returning back to the airport to do some landings. The student pilot crossed the airport at mid field and entered left traffic following another landing airplane. The student pilot made a normal landing to runway 24. The CFI felt the pilot apply some brakes and informed the student pilot to maintain directional control with the rudder only. The student pilot complied with his instructions, taxied the airplane to the end of runway 24, and turned left onto the taxiway. The other airplane was on the parallel to the runway 24 taxiway. The CFI instructed the student pilot to stop the airplane after clearing the runway. The airplane went further than the CFI wanted it to go and he told the student pilot to apply brakes. The CFI applied brakes and the airplane would not stop. The CFI applied the emergency brake and the airplane would not stop. The CFI pulled the mixture to the cut off position and attempted to stop the airplane on the lip of the edge of the taxiway. The airplane did not stop and went down an embankment and collided with trees. The registered owner/operator informed the FAA that the brake system had been serviced one week before the accident by adding brake fluid. Review of the airframe logbook revealed no entry had been entered into the logbook for the servicing of the brake system. Two FAA inspectors examined the airplane on March 23, 2007. A functional test of the brake system was performed by pushing the airplane with the parking brake engaged, "which indicated a braking system failure." Additionally, a similar functional test was performed with one inspector in the airplane applying the brakes from the left seat, which resulted in a brake failure. The test was performed from the right seat with identical results. The brake pads and rotors were visually inspected and no anomalies were noted. "Visual inspection of the master cylinders indicated that brake fluid was leaking onto the exterior of the master cylinder components evident by the staining of dried hydraulic fluid. During the same inspection, the carpet near the pilot's (left front seat) rudder pedals were saturated with brake fluid. There were indications of some recently spilled brake fluid (which had a reddish color) and brake fluid that had spilled some time ago (which was brownish in color)."