RPTR AND FLT INSTRUCTOR RAN OFF SIDE OF RWY ON LNDG. RWY EXCURSION.

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EVERYTHING PROPERLY SET UP FOR A SAFE LNDG; OUR AIRSPD WAS 50-55 KTS ON TOUCHDOWN. AS SOON AS THE ACFT TOUCHED DOWN; IT BEGAN TO PULL SHARPLY TO THE L; EVEN THOUGH R BRAKE WAS FULLY APPLIED. THE AIRPLANE WAS ALREADY OFF THE RWY WHEN MY INSTRUCTOR HAD FULL CTL. THE GND WAS EXTREMELY SOFT AND AS THE NOSEWHEEL DUG INTO THE GND; THE PROP ALSO DUG INTO THE GND. THE CAUSE OF THIS INCIDENT IS UNDOUBTEDLY A PROB WITH THE BRAKE; THE AIRPLANE WAS ON THE CTRLINE UPON TOUCHDOWN AND ALL POSSIBLE ACTIONS WERE TAKEN TO COMPLETE A SAFE LNDG. SUPPLEMENTAL INFO FROM ACN 299726: AFTER A LOW APCH (NDB-GAMN) THE PVT PLT RECEIVING INSTRUMENT INSTRUCTION; BEGAN THE SDF-9 (AMN). THE APCH PROCEEDED WITHOUT INCIDENT. AFTER FLYING THE APCH TO MINIMUMS THE LNDG WAS ATTEMPTED. AT APPROX THE HALFWAY POINT OF THE RWY THE CESSNA 172 MAIN GEAR TOUCHED DOWN WITHIN 7 FT OF THE CTRLINE ON THE S SIDE (R). (5000 FT LONG; 75 FT WIDE; HARD SURFACE). PVT PLT HAD THE CORRECT XWIND INPUTS APPLIED. UPON TAKING CTL OF THE ACFT; I APPLIED R BRAKE STRONGLY. THE BRAKING ACTION COMBINED WITH R AILERON WAS NOT EFFECTIVE IN STOPPING THE L TURN. THE ACFT CROSSED OFF THE RWY; WHERE THE GND SLOPED DOWN AND TO THE L (N) FOR DRAINAGE. AS THE ACFT PROCEEDED DOWN THE SLOPE THE NOSEWHEEL BEGAN TO DIG INTO THE SOFT GND. WITH THE ACFT NOSEWHEEL DIGGING IN AND THE ACFT BANKED SLIGHTLY TO THE L (FROM THE SLOPE); THE PROP STRUCK THE GND ALONG WITH THE L WINGTIP. THE ACFT STOPPED WITH THE NOSEWHEEL DUG INTO THE GND. I BELIEVE THE CAUSE OF THE ACCIDENT WAS A STUCK L BRAKE AND/OR THE PVT PLT (MANIPULATOR OF THE CTLS) LANDED WITH THE L BRAKE ON. THE ACFT WOULD NOT HAVE STRUCK THE GND (PROP-WINGTIP) IF THE GND WAS HARDER. TO AVOID THE ACCIDENT SHOULD HAVE BEEN KEPT ON THE RWY. IN MY OPINION THIS WAS IMPOSSIBLE WITH THE TIME/SPACE AND STUCK BRAKE WORKING AGAINST ME.