Flight Instructor with student reported a runway excursion resulted when the student conducted a high-speed taxi after landing to exit the runway before another aircraft landed. Instructor taxied aircraft back to the ramp.

Narrative

The event occurred while we were conducting a stage check. We had flown out to the practice area; with the intention of conducting the landings at the ZZZ airfield. We entered the pattern with two other planes; who were also using the XX Runway. While conducting Power-Off 180 landings we executed one; determined we floated too long; and took off to do a second attempt. During the second attempt we conducted a go around and decided to try for a third time. I informed the student at this time that we would do the landing to a full stop so as to conduct a short field take off. She managed to land on this attempt however forgot that I told her full stop; so added power for a second before I corrected her. This led us to traveling further down the runway than initially planned. Based on this I determined that we didn't have enough runway left to conduct the take off and instructed her to taxi down to the end of the runway and taxi back for Runway XX. While she was conducting this; we had another plane coming in to land and so she added more power to try and speed things up. I was somewhat distracted thinking about the landing and not being diligent enough in observing her. I noticed that we were going a bit fast too late and we approached the end of the runway without her being able to fully stop. I jumped on the brakes with her and added right rudder (in the direction of the taxiway). We departed the runway with (what we later learned) all 3 tires. However; we managed to not hit any of the lights or other obstacles present in the process of making that turn. Due to the nature of going off the runway but not feeling like we hit anything; I decided to taxi back to the ramp; shut down and inspect the plane and call the Chief Pilot and our Maintenance Department about the next steps to ensure the aircraft was undamaged and we were safe to fly in again. Cause: The rush to get off the runway due to incoming traffic; stage check nerves; possible seat position of the student; and a lack of vigilance on my part as the CFI. Suggestions: Ensure that my attention; especially with a student I hadn't flown with; was not distracted at all. I would also check that she could reach all of the controls effectively; so that she could exert full force on the brakes. I would also not allow individuals to taxi (even on the runway) that quickly; even with a plane coming in behind us; they could always conduct a go-around. And finally; I would be certain to keep monitoring the student's control inputs so that I could hopefully stop any inappropriate control inputs.