C172 Flight Instructor reported a tire failure on landing during a training flight; resulting in a runway excursion. The pilots evacuated and there was no damage to the aircraft.

Narrative

Flying with student and practicing short field performance landings. 4th landing of the lesson; all prior landings had been within standards and safe; student was not riding the breaks. Inbound to ZZZ from ZZZ1 winds reported calm we entered 45 to the left downwind for RWY XX. After one landing we did a full stop and changed to RWY XY for me lightly favorable winds. We continued the lesson on RWY XY. Downwind for RWY XY; incoming traffic made an initial call for left base XY so we did a right 360 for spacing as we were the only traffic in the pattern. Re-entered the downwind and continued descent and configuration. Student was holding appropriate descent and airspeed for the short field landing. We touched down aligned with centerline and at appropriate airspeed (68 mph); no slamming or porpoising. Upon landing the aircraft started to slew towards the left. My feet were on the pedals; both myself and the student began to apply right rudder and right brake to correct the leftward motion. The aircraft continued to the left of centerline before exiting the runway and stopping in the grass. Neither myself or the student was harmed; upon exiting we saw the tire was flat and skid marks on the runway where the tire started to dig into the pavement. Upon visual inspection there was no exterior damage and no prop strike.