A Cirrus SR-20 pilot reported brake failure during landing resulted in a loss of control and runway excursion.

Narrative

I was doing a discovery flight with a brand new student with no flight experience. I had flown a lot in another G3 SR20 but this was my first flight in this specific plane. I noted that the brakes felt different on the preflight check but they held for the start up and the run up so I thought maybe it was just something quirky about the plane. I was flying on the right as the CFI. We went out and did some manuevers; very routine discovery flight. the student was performing quite well so when we turned to come back into the class C airspace I allowed him to continue flying. I took the controls back about 4-5 miles from the runway. At that point the wind had picked up a bit more than I was expecting. I did my prelanding flow but did not pick up the paper checklist to confirm that I checked the brakes. I cannot remember if I checked pedal pressure for both; But I do remember that I did not check the brakes individually; left/ right. I landed with a bit more speed than normal because of the crosswind and the gusty nature of the wind. I typically slow down with aerodynamic braking first before putting too much pressure on the toe brakes. I did note however that it was taking me much longer to slow down than normal. ATC told me to turn right at the next taxiway and when I slowed to what I felt was a safe taxi speed I initiated the turn to the right with the brake. My foot went all the way to the floor; allowing the rudder to turn me but I wasn't getting any toe brake pressure so I did not turn enough. I kept pressing the right brake to try and get it to come around and it did not; so we went into the grass. My human performance was first compromised by the shock of not having a response from my right brake. It took me too long to ask the student to try his. This was also a scenario I had never rehearsed on been trained on during any of my flight training or check rides; so there was no reflex there. Thankfully we had just a small scratch on the plane and we were unharmed but this is a scenario I have now replayed in my mind many times. I now wish I had trained on this more especially in planes that are steered with the toe brake. I feel this needs to be addressed in flight training.