PA-28 Flight Instructor reported engine failure during training flight. Instructor requested and was provided priority handling. Aircraft was unable to reach runway and resorted to safe forced landing on a nearby field. No injuries or aircraft damage reported.

Narrative

On Date; aircraft X experienced a full power loss which led to an off airport landing approximately four miles southeast of ZZZ1. The aircraft was inspected by student and instructor prior to flight. The pre-flight inspection showed that the aircraft appeared to be in normal working condition with no visible defects. The fuel tanks were visually inspected to be full prior to flight. The fuel was drained and inspected from all three fuel drains and did not show any signs of water or other contaminants. Engine run up was completed with no signs of abnormalities. The aircraft took off on Runway XX from ZZZ1 and turned to the east. Name flew eastbound approximately 22 miles to identify checkpoints for a practice cross country. Then; to practice diverting to another airport; he flew south toward ZZZ2. About 15 miles north of ZZZ2; the diversion procedure was deemed satisfactory; the student proceeded west/southwest to practice lazy 8 maneuvers. The electric fuel pump was turned on and the fuel selector was switched around this time from the left tank to the right. No interruption to power or any other engine indications was observed. The electric fuel pump was then turned off per Piper procedures. After three satisfactory lazy 8 maneuvers were performed; the instructor slowly pulled the throttle to idle to simulate an emergency power off landing. The student then proceeded to simulate what he would do in this situation.No system (magnetos; electric fuel pump; mixture; fuel selector) was actually touched during this time; the student simply verbally acknowledged what he would do in this situation while the throttle remained at idle. After the simulated emergency was satisfactory; the aircraft climbed back to approximately 2000 feet and began proceeding direct to ZZZ1 to practice landings. The engine produced a loud 'pop' noise that sounded similar to backfiring; followed by a slight decrease in RPM. At this time the instructor took control of the aircraft and began troubleshooting the aircraft.The instructor turned on the electric fuel pump; instructed the student to switch the fuel tank from the right to the left; and turned on the alternate air. The alternate air provided a slight and momentary increase in RPM. However; it did not last long and the popping from the engine continued to get worse and more frequent. The RPM also continued to steadily decrease each time the engine popped as well. The magnetos were also cycled during this time; however; no remedy attempted by the instructor made any difference in performance for the engine other than the momentary and slight increase of power during the initial application of alternate air.An initial [request for priority] was made by the instructor during this time to ZZZ1 tower for a partial power loss and engine roughness. The tower immediately cleared aircraft X to land on runway XXL. The aircraft proceeded directly toward runway XXL while performance was slowly continuing to decrease. One more radio call was made shortly after that indicating that aircraft X may completely lose the engine upon touchdown due to continuously degrading engine performance and Tower acknowledged. As the engine performance continued to steadily decrease with no sign of relief; the instructor became aware that making it to a runway at ZZZ1 was becoming extremely unlikely. The power had continuously and steadily decreased over the course of 6 to 7 minutes. However; as soon as the RPM reached approximately 1900 RPM the entire engine completely stopped. Throttle was entirely unresponsive.A landing field was then chosen approximately 3.5 miles southeast of ZZZ1 and the instructor maneuvered to land in the field. While turning toward the field; one last call was made to tower to advise that the engine had completely failed and that we were going to land in a field. Due to the little amount of time between the complete engine failure and touchdown; full engine securing procedures were not able to be accomplished as the pilot was focused on maneuvering the aircraft safely to touchdown.After the aircraft touched down and came to a complete stop; engine securing procedures were accomplished. The pilots assessed that there was no damage to the aircraft during the landing and no injuries sustained. Local EMS arrived shortly after touchdown to assist and gather information. The chief pilot and head maintenance supervisor arrived shortly thereafter to assist as well. These statements have been recollected to the best of my knowledge and have been retold as accurately as I can possibly describe them.