PA34 INSTRUCTOR AND STUDENT EXPERIENCE RWY EXCURSION AFTER A BRAKE FAILURE.

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I WAS THE MULTI-ENG FLT INSTRUCTOR ON A FLT TRAINING MISSION IN A PIPER SENECA (PA34-200); WHICH HAD JUST COMPLETED A 100 HR INSPECTION 2 DAYS PRIOR. THIS PARTICULAR SENECA DOES NOT HAVE TOE-BRAKES FOR THE R SEAT PLT (INSTRUCTOR'S POS). THE INSTRUCTOR MUST RELY UPON AN EMER HAND BRAKE SHOULD THE STUDENT PLT NEED ASSISTANCE. WHILE FLYING THE ACFT ON THE DOWNWIND LEG OF THE ARPT TFC PATTERN; THE PRE-LNDG CHK WAS COMPLETED; AND PROPER BRAKE PRESSURE WAS VERIFIED BY THE STUDENT. THE ACFT WAS LANDED BY THE STUDENT WITHOUT INCIDENT ON THE SHORTER OF THE 2 RWYS. THE STUDENT WAS HAVING DIFFICULTY TRACKING THE CTRLINE OF THE RWY; HOWEVER HE DID NOT STATE THAT HE WAS HAVING BRAKING PROBS; XWINDS WERE ALSO PRESENT; AND SO DEVS FROM THE RWY CTRLINE APPEARED TO BE WIND-RELATED. AT APPROX 30 MPH THE STUDENT TOLD ME HE WAS HAVING PROBS BRAKING; AND THE AIRPLANE WAS APCHING THE END OF THE RWY. THE AIRPLANE'S R TOE BRAKE WAS 'SPONGY' AND NOT WORKING; HOWEVER; THE L BRAKE WAS WORKING. AS HE DEPRESSED THE BRAKES; THE AIRPLANE VEERED OFF TO THE L OF THE CTRLINE. MY CTL INPUT FROM THE R SEAT (WITH NO BRAKES) WAS HVY R RUDDER TO COMPENSATE FOR THE INDUCED L DEV CAUSED BY THE STUDENT'S BRAKING. I SAID TO THE STUDENT 'MY AIRPLANE' AND PULLED THE HANDBRAKE; WHILE CONTINUING TO APPLY R RUDDER SO AS NOT TO RUN OFF THE L SIDE OF THE RWY. THE AIRPLANE VEERED HARD TO THE R WITH HAND BRAKING. THE AIRPLANE THEN LEFT THE RWY ON THE R SIDE AT APPROX 20 MPH; CROSSED A FLAT AREA WITH WEEDS; AND ENTERED AN OPEN PARKING LOT AREA WITHOUT STRIKING ANYTHING. NO DAMAGE WAS DONE TO ARPT PROPERTY OR THE AIRPLANE. I WAS HANDED OFF TO GND CTL; AND I TAXIED THE AIRPLANE VERY SLOWLY WITH HAND-BRAKING ROUGHLY 50 FT TO CLR THE ARPT MOVEMENT AREA. SHORTLY AFTER; THE R BRAKING SYS THAT HAD FAILED EARLIER WAS WORKING FINE; AS IT SEEMED TO HAVE BEEN DURING THE BRAKE CHK ON THE DOWNWIND LEG.