PA-28 Student Pilot reported the engine malfunctioned and eventually failed during a training flight. The Flight Instructor requested priority handling and attempted to return to the airport but was unable to reach the runway and instead safely landed on a field away from the airport.

Narrative

I am a student who has a Private Pilot License with an instrument rating. The instructor and I took off out of ZZZ around XA:10 to conduct my stage check. We first flew eastbound passing ZZZ1 at 2;500 ft. MSL and reaching around the town of ZZZ2 at 3;500 ft. MSL to do our cross-country portion. Once we reached ZZZ2; the instructor told me to divert to ZZZ3. So; I turned southbound to demonstrate a simulated diversion. Once we were about 1/3 way to ZZZ3 from ZZZ2; he said he was satisfied and wanted me to show him Lazy 8s.I demonstrated my Lazy 8s and he told me to take him towards the airport. So; I turned westbound and reached an area between ZZZ4 and ZZZ5. Once we were at that area; the instructor created a simulated engine failure scenario by putting the throttle to idle. I maintained my best glide speed; picked a field; and did simulated engine troubleshooting and restart - didn't actually turn on/off any buttons or controls; I just verbalized because it was just a simulation to demonstrate my knowledge. Once we got low enough; he was satisfied; so he told me to recover. I executed a go-around and leveled off at 2;000 ft. MSL at around 2;300 - 2;400 RPM. From there; I flew northbound towards the town of ZZZ6 because he wanted me to take us back to ZZZ to do [a] couple [of] landings; which were the last item that I was supposed to be tested on. While I was flying towards ZZZ6; I got the weather information from ZZZ ATIS. Once we were south of the ZZZ6; the engine made a loud pop sound and began to drop RPM. It stayed around 2;000 RPM; 90 kt.; and 2;000 ft. MSL.From there; the instructor took the controls. We were troubleshooting the engine and once he opened the alternate air; the RPM was regained slightly and the engine seemed to be doing better. However; that RPM relief was short-lived; and the loud pop noises became more frequent and the RPM and the altitude were slowly dropping. At that point; we [requested priority handling] to ZZZ Tower that we are experiencing partial engine loss and that we need to land as soon as possible. The Tower acknowledged our [request] and told us to take whichever runway; and we said we are going to take Runway XXL.We were making our way to the runway. However; all of a sudden; we experienced a complete engine failure from a partial engine loss. At this point; we were north of ZZZ6; about 5 - 6 NM south from ZZZ; around 1;400 - 1;500 ft. MSL. He told me to squawk 7700; which I did. We also tried to restart the engine but the engine did not restart; so we secured the engine and committed to landing on the field because we were definitely not going to make the runway. Our decision was correct; because the instructor was able to safely land the aircraft; Aircraft X; on the field by ZZZ7 and ZZZ8.The Tower asked us if we were okay and asked for our location; so they can send emergency service to us. We gave them our location by using our iPhones; and the fire trucks and EMS truck arrived shortly after. We completely shut the aircraft down. Thankfully; the instructor and I did not sustain any injuries; and there were no damages to the aircraft other than the failed engine. Soon after; our Chief Flight Instructor and Chief Maintenance Staff came to the scene. They tried to troubleshoot the engine to figure out what was wrong with the engine; but they couldn't come up with a concrete answer. Therefore; they said they will send more equipment and maintenance crew to check out the aircraft. The Chief Flight Instructor talked to the fire fighters and the property manager; and then Chief Flight Instructor; the Chief Maintenance Staff; the instructor; and I left the scene and came back to ZZZ. The instructor and I debriefed; and I left the airport to go back home.