C172S Flight Instructor reported conflicting traffic failed to go around safely as announced resulting in evasive action a conflict.

Narrative

Situation invloves Aircraft X and Aircraft Y in the pattern at ZZZ. Aircraft X on final. Aircraft Y gave 1 call at 3 miles and enter line left downwind. Had us (Aircraft X) in sight on final. Aircraft Y announced base very close to us; Instructor Pilot (IP) said stop and go; to which Aircraft Y reported he was going around. We continue approach and land. On landing roll to stop; IP noticed Aircraft Y was never going around and about to land. Aircraft Y yelled 'come on Aircraft X; hurry up.' about 100ft high above runway; IP takes controls to takeoff to avoid traffic that was landing and not going around. IP realized due to airspeed; we did not have enough time to come to stop on runway; so IP elected to takeoff. Stall horn coming on at 41 KIAS. IP went right into ground effect to build speed and climbout thankfully clearing trees with not much time left. Student noted; we started takeoff roll past grass strip approaching Runway XX at ZZZ. Still in ground effect past threshold to get enough speed to clear trees. Student noted we rotated at 41 KIAS and got to 51-52KIAS in ground effect before climbing out and clearing trees by less than 50 ft. IP elected to discontinue from EOC (End of Course) due to situation. IP said on frequency that 'you said you were going around; you need to do that next time.' IP does not recall what Aircraft Y said back. IP responded with 'you need to do better.' Aircraft Y responded with 'I have 50 years of experience in aviation; what do you have.' IP did not respond back to Aircraft Y and left the pattern and returned safely to ZZZ1. *Better communication could have been shown here to help prevent situation which seemed to be caused from lack of communication. *IP of Aircraft X could have noticed ASI sooner which could have led to full braking on runway and getting off quickly instead of taking back off to clear obstacles.