Flight Instructor with student reported several near misses with an aircraft that appeared to be maneuvering in the same area.

Narrative

This report concerns a dual instructional flight conducted in Aircraft X with one student. We departed ZZZ at approximately XA30 and proceeded to the Practice Area; which begins approximately 10 to 15 NM southeast of the airport. Once in the area; we climbed to 3;000 feet MSL and began working on performance maneuvers. At various points; we climbed as high as approximately 3;700 feet MSL. I cannot recall the exact maneuver we had just completed; but sometime between XB30 and XB50; we transitioned into power-on and power-off stalls near the town.As my student initiated the first stall; our ADS-B IN traffic alert system issued a 'TRAFFIC' alert. I looked outside and saw nothing; then checked the MFD and noticed a traffic icon below and slightly behind us. I had my student recover early from the stall so I could evaluate the situation. The traffic icon disappeared shortly after; which partially supported my suspicion that it was another GPS 'ghost.' We have experienced ghosting in these aircraft before during steep turns; but this happened while setting up for a stall.I mentioned this to my student and noted that the situation felt different. To see if it was a GPS issue specific to our location; I had her climb and reposition us slightly away from the area. The idea was to see if a change in location would clear up any anomalies. Once repositioned; I had her attempt a second stall.As she began the setup; the ADS-B traffic again appeared; gave an alert; and disappeared. This did not fit the ghosting behavior we've seen in the past. We received another close-range alert during the maneuver. I had her recover again; and at that point; I began seriously questioning whether we were dealing with ghosting or an actual aircraft.I contacted ZZZ Approach on XXX.XX and requested flight following. I asked if they saw any traffic in our area; and they said they did not see anything at that moment. Just after she said that; the aircraft climbed high enough just off our right side that I made visual contact. We visually acquired the aircraft at our 3 o'clock. It appeared to be a white; low-wing airplane; moving fast and close enough that we could see it fairly clearly; but not so close laterally that I could tell whether it was single- or twin-engine. That was the moment I took control of the aircraft.A short time after that; the controller said the aircraft had popped up on their radar; but the signal appeared to be intermittent. They suspected a malfunctioning transponder.My student and I both began to feel uneasy. We realized the alerts we had been getting may have been actual near misses. The aircraft passed quickly out of view; but the speed and proximity were enough to rattle us. ZZZ ATC later reported it was moving at approximately 160 knots. I became concerned that the aircraft was trying to come up under us or was unaware of our presence. I performed a chandelle to gain altitude and turn away.At some point during this exchange; I told ATC that the aircraft appeared to be within 50 to 100 feet of us in altitude at times. Looking back; I now realize I misinterpreted what I saw. The ADS-B showed '-5;' which I stated as 'probably 50 feet;' but in hindsight I know that meant the traffic was 500 feet below us.In the time following; I began relocating us toward the local practice areas. I accidentally referred to it as the other area when speaking with ATC; but our intention was to shift to a different part of the local practice airspace.My memory of exact timing and altitude is a bit unclear at this point. The events were unfolding quickly; and I was focused on avoiding the threat. At one point; I saw either the aircraft itself or its ADS-B return again. It seemed to be close to our altitude and possibly next to us. From that point forward; we also began receiving additional advisories from ATC regarding the aircraft's location; including times when it was near us and when it appeared to be following us. Thinking quickly;I performed a forward slip to gain both lateral and vertical separation; descending and increasing speed to get out of the area.We felt safer afterward; but both my student and I were visibly shaking. We agreed that we were no longer 'IMSAFE' and decided to terminate the lesson and return to ZZZ to land. ZZZ Approach asked for our intentions; and I advised them that we were inbound to ZZZ. They warned us that the same aircraft still appeared to be trailing us near Location A. ZZZ Approach then canceled our flight following so we could transition to CTAF.After making my initial call to ZZZ traffic; I heard another voice respond. I thought I heard them mention 'RV;' which I interpreted as the aircraft type. I asked if they had just been over Location A. I do not recall saying that I was in contact with ATC; but I do remember telling them directly; 'You guys need to contact ZZZ ATC.' The pilot then asked if we wanted them to break off and said they did not mean to scare us. I responded; 'Too late for that; sir.' That was not a professional comment on my part; and I regret saying it. The aircraft then departed the area; heading east of the airport and did not attempt to enter the pattern.After landing; I spoke with my Chief Flight Instructor and later contacted ZZZ ATC. I thought I was speaking to the same controller from the flight; but it was actually the ATC manager. She asked for the details and explained that radar coverage did not begin until we established flight following; so the first several alerts and possible near misses would not show on their records. She suggested the other aircraft may have been practicing ground reference maneuvers and that it could have been coincidence we were in the same place.That made me pause. It is possible this was all an unfortunate set of coincidences. If so; I do not want to wrongly accuse another pilot. I hesitated to file this report for that reason. But I still find it hard to believe this was all unintentional. I continue to wonder: - How did the other pilot not see us or avoid us during multiple close passes? - Why did they appear to follow us back toward the airport? - If they were returning to land; why did they break off and fly away once I called them out over the radio?If this truly was a case of bad timing; poor equipment; and shared airspace; then it is a textbook example of the Swiss Cheese Model. Two similar aircraft in the same practice area; different frequencies; one with a possible transponder failure; one relying on ADS-B; a rattled student and flight instructor; and increasing uncertainty. But if the other pilot was intentionally flying close to us; then the situation is much more serious.In hindsight; I made my own mistakes. I hesitated too long; hoping it was just a GPS issue. I let fear and emotion creep into my judgment. I also could have made a CTAF position call sooner; before reaching out to ATC. Listening back to the audio; I noticed that my radio calls came across more casual than they should have been. I believe this was due to nerves at the time. I did not speak the way a professional pilot should over the radio during most of the flight.In my experience; I regularly fly in close proximity to other aircraft in structured and legal operations; such as during collegiate flight team events or while towing gliders; including in contest environments. I am not easily rattled by aircraft in my vicinity when I am aware of them. But this was different. The repeated; uncoordinated proximity and lack of awareness from the other aircraft surprised me and had a serious effect on both my student and me.Our ADS-B in and out were functioning properly. ZZZ ATC has radar logs from the time we were on flight following; and third-party sources like ADS-B Exchange may also have data.I hesitated to file this because I was unsure whether I was overreacting. But even if it turns out to be a chain of coincidences; I believe this incident should be reported.It is worth documenting for safety; training; and awareness purposes.