DA-40 flight instructor reported a conflict during their landing roll with an aircraft that was on final for the same runway. The landing aircraft was instructed to execute a go-around.

Narrative

I as and instructor and a student were flying a Diamond Star (DA-40) on a private pilot training flight. In right traffic ATC gave clearance 'Clear for the option RWY XX' to which my student responded 'Cleared for the option XX (Our tail number)'. Shortly after our clearance was received it was noticed that the controllers had now switched. We proceeded down to land. After landing; we proceeded to slow down; unable to make Taxiway 1 at ZZZ due to speed. We continued to slow our speed and move down the runway. At this time I requested 'Tower (our tail number) request turn onto [Runway] Z/AA 'to make the landing a full stop and indicate that to the tower; as allowed per our 'Cleared for the option' clearance. We received no response and at this point had slowed to a relatively low ground speed of 5 to 10 knots but continued down the runway at that speed anticipating further instructions due to no remaining taxiway turnoffs until the end of the runway. We then received a transmission from the tower stating 'Do not stop on the runway; Do not stop on the runway' before he advised traffic on final to Go Around. At this point we were at a low groundspeed between; however our aircraft was not stopped on the runway as has been stated. We continued our 5-10 knot rollout to assure proper aircraft control in the training environment; never stopping on the runway and rolling just past the intersection of Runway XX/YY and Z/AA. Seconds later ATC then advised '(our tail number) possible pilot deviation; do a 180 on the runway back taxi Runway XX; [Taxiway] 1; [Taxiway] 2 to parking' to which I read back. I asked ATC 'Was I cleared for anything different than previously' to which ATC advised 'No you were not; which means turn on a taxiway or continue to the end'. We proceeded to follow our taxi instructions back to the ramp with no further occurrences. I was not given a phone number from the tower to contact them after the flight. When being cleared for the option; as the pilots we understood that we had the option to use the runway to land full stop; touch and go; stop and go; low approach or missed approach at the discretion of the pilot.