Flight Instructor reported a student's runway excursion due to excessive landing speed which resulted in a loss of positive aircraft control on runway. Flight Instructor returned onto the runway and reported no aircraft or airport damage.

Narrative

My pre-solo student was coming in to land after one prior landing and trip around the pattern. Their approach was stable down final and I was shadowing the controls to assist with the landing. As the plane flew over the threshold; he pulled the power to idle and allowed the plane to continue descending toward the runway surface. I noticed that my student was not initiating their flare; so I instructed them to add aft elevator pressure and flare. When my student did not comply; I assisted him on the controls. We touched down with a higher speed than normal; my student did not have positive control of the plane. I called for the controls twice before they released the controls; at which point the airplane was fishtailing on the runway. In that moment; I made the decision to neutralize the rudder pedals and allow the airplane to travel toward to the left side of the runway; instead of attempting to swerve the airplane on the runway while not having positive control of the movement of the airplane. I was able to apply brake pressure and hold aft elevator to slow the airplane down; but could not stop before the airplane exited the left side of the runway. When it became inevitable that the airplane was going to exit the side of the runway; I made it my first priority to regain positive control of the airplane; and slow it down so that I could ensure that we did not hit a runway edge light; or damage the gear on the airplane. When the airplane exited the runway; I held aft elevator pressure to protect the nose wheel; and the airplane had slowed enough to allow me to turn back onto the runway. I immediately contacted tower and taxied the airplane onto nearest the taxiway. The controller responded by stating they saw the event; asked if the airplane was damaged or if we had hit a taxiway light. I responded by saying we landed with too much speed and lost positive control of the airplane; but did not hit anything or cause damage to the airplane. The controller responded to my transition by saying 'okay; contact ground on frequency'; and that was the end of the event. I believe that upon touchdown my student froze on the controls; which restricted my ability to control the airplane. Immediately when I felt the situation could become unsafe; I called for the controls; but my student did not let go of them. It was at this point after touchdown that the airplane began to fishtail and we lost positive control. I called for the controls again; at which point my student let go; and I felt that the safest option was to neutralize the rudder pedals and allow the airplane to travel in a straight line toward the side of the runway; while I attempted to slow it down. Immediately after we had slowed to a speed that was safe enough to turn the airplane; I turned back onto the runway and contacted tower. After debriefing the event with my student; I believe we both understand why the situation unfolded in that manner. Moving forward; I am confident that when I call for the controls if I feel as though I need to take control of the situation; I will have sole manipulation of the controls; and be able to carry out the flight safely.