DA40 Flight Instructor reported aircraft had a hard landing and or return to airport loss of brake control resulted in runway excursion.

Narrative

My student (flying right seat) and I were completing Lesson 18 in the CPL 141 program for flight school. Student had been performing the soft field and short field landings and takeoff's with great skill so we jumped ahead and started working on Power off 180's (actually demonstrated in lesson 16 and he was flying them in lesson 17 as well). We had been as ZZZ1 for about an hour working on landing when the hard landing occurred.Student had turned final on a power off 180 attempt and was on glide slope with a noted airspeed of 62/63 kts. As we approached the threshold of the runway (about 75 ft. AGL) the plane started to sink at an increased rate. Student performed two actions at this time. The first was to pitch the nose down to increase airspeed. The second was to decrease flaps from 30 degrees (full flaps) to 0 degrees (no flaps). The plane; already pitched forward to increase airspeed; then dropped when the flaps were retracted. I grabbed the controls at this time and pulled the nose up as we were nose down straight at the runway attempting to soften the impact thinking to myself that if I added power it would just put us into the runway at a faster airspeed. I got the nose up before we impacted but the plane still; for lack of a better term; pancaked into the runway. Upon impact; the plane bounced back into the air. I immediately added power to avoid porpoising and then we performed a go-around.I noticed no funny noises or vibrations during the go-around and the plane flew normal with no abnormalities noted in flight. Student then performed a soft field touch and go with no issues. We then left the pattern and headed west of ZZZ1 where Student performed slow flight; a power off stall; a power on stall; demonstrated accelerated stall then a steep turn. I could tell he was not feeling well and upset about the hard landing so I asked him if we should call it a day and had back to ZZZ to which he agreed.We flew back to ZZZ and landed on Runway XX; again; no abnormalities noted during the flight back. Upon landing; another soft field landing without incident; we were instructed to taxi down to Taxiway X to get off the runway. When we got close and needed to start braking; Student mentioned to me that he had no brakes. I took controls of the plane and noticed I also had no brakes. We rolled past Taxiway X and I called the Tower letting him know we didn't have any brakes. I then serpentined down the runway with the power out slowing us down until we reached the end of the runway. We turned off the engine and slowly rolled off the end of the runway onto the dirt and gravel where the Tower had a tug come and then pull us back to the hangar.