C177RG flight instructor reported a malfunction in the brake system during taxi for takeoff.

Narrative

I was providing flight instruction to student in Cessna 177RG. The weather was VFR with light winds from the south and temperatures in the 40's F. The student has a current/valid FAA Private Pilots certificate. The lesson began on the non-movement ramp of Airport FBO. The aircraft was towed from the T hangar to the ramp by the FBO personnel and parked facing north on the south side of Taxiway XX. I began the lesson with a preflight briefing in the FBO lobby and covered the plans for the day. The student appeared well rested and healthy and showed no signs of cognitive duress. The student understood the objectives and the parameters for the lesson; and we proceeded to the aircraft. Upon reaching the aircraft I observed the student complete a thorough preflight inspection of the aircraft which included all systems. I also visually inspected the aircraft and noticed no signs of undue wear and tear or malfunctioning systems. Inside the aircraft the student completed the cockpit prep checklist and started the engine with no sign of issue. Although cold; the engine started immediately and setup for our lesson continued. Student completed his preflight briefing and confirmed the aircraft was ready for taxi. On initial contact the controller issues taxi instruction to Runway XX via Taxiway XX and XY. Student released the parking brake and began the taxi roll. No issues were observed with the left turn onto Taxiway E. The student taxied the aircraft towards Taxiway XY and reduced the speed with power in preparation for the right turn onto Taxiway XY. As the right turn was initiated; I noted the student began to turn the yoke in preparation for the quartering tailwind on the northbound taxi. As the aircraft failed to make the turn and continue due west towards the taxiway edge and grassy area between Taxiway XY and Runway XX; the student said; 'something is wrong' and pulled throttle lever as far to the aft as it could travel. I noted at that point an issue with the aircraft and attempted from the right side of the aircraft to bring it to a stop. I noted immediately the right brake pedal depressed fully forward with no resistance against my foot. Due to the attempts from both pilots to stop the aircraft; a hard left turn was initiated back to the south and in the turn; the prop struck a blue taxiway edge light. The aircraft came to rest facing south with the tail hanging over the west taxiway edge line and the prop facing the center line of the taxiway. I assured that both myself and the student were uninjured in the event and notified the Ground Controller of our incident. Ground Control advised that if able we were clear to taxi back to the non-movement area and I complied with that instruction without further incident. I observed during that short taxi that the right brake pedal had no resistance and therefore no braking ability. An Airport Operations vehicle was in the area and observed the incident occur and immediately began to clear the taxiway of debris from the light that was struck.A post-incident inspection of the aircraft showed no visible issue with the aircraft until I laid down onto ground and slid myself completely under the aircraft. At that point I could see a significant amount of brown fluid on the underside of the engine block and beginning to run down the backside of the nose wheel landing gear system. As I was coming back from under aircraft; I also noticed damage to the tip of the prop from the taxiway edge light being struck.The student was able to get a phone number for Airport Operations to ensure they were aware of the incident involving the taxiway edge light. I completed a debrief of the incident with the student and we left the aircraft in the non-movement area of the FBO ramp.