A Flight Instructor reported the engine lost power on initial climb out and a return to the departure airport.

Narrative

I was accompanying another pilot and providing instruction for a newly-purchased PA-28-235. We were intending to depart ZZZ1 and bring the plane to its new home at ZZZ2; with a planned fuel stop at ZZZ. No issues were noted during my post-maintenance flight nor on the first leg from ZZZ1 to ZZZ. After an hour on the ground to refuel; we had a normal preflight and engine runup for our second leg ZZZ-ZZZ2. I was in the right seat and the owner was flying from the left seat. Takeoff from Runway XX was uneventful; but at approximately 300 ft AGL I noticed the propeller RPM went above redline to around 2850. I mentioned this and brought the propeller lever back to try and reduce RPM. This had no effect and at that point I announced that I had the flight controls. At this point; CHTs were getting very hot; above 425F. Oil pressure was normal. I reduced throttle to keep the RPM below redline. This required a fairly low power setting which; combined with a hot day and higher weight; meant we were barely climbing. I decided our best option was to return to the departure airport. Knowing the winds were calm; I elected to turn back and land on Runway XY. There was one other aircraft in the pattern who had just taken off behind us. I announced on CTAF that we had engine problems and were coming back to land on Runway XY. The other aircraft had already taken off and was above our altitude. I made an uneventful landing and taxied back to the FBO. We removed the cowl and saw the governor was still connected and moving the propeller lever actuated the part. After consulting with maintenance; we ran the engine up and tried reducing RPM with the prop lever at various power settings. This had no effect and we suspect the governor itself has failed. There were no signs of oil leakage.