Flight instructor reported taking evasive action several times in the traffic pattern at non-towered airport GVL to avoid another aircraft that cut them off and was following behind the instructor's aircraft very closely.

Narrative

I was doing a takeoffs and landings training flight at an GVL (untowered) turning onto the downwind when Aircraft Y makes radio call looking for traffic advisories. At this point I saw him on ADS-B appearing to be approximately 5 miles out. As I turn onto left base and make my CTAF call; he snuck up right behind me and turned base seconds after me. At this point I communicated that he was cutting me off and I began a turn right to re-enter the downwind and avoid him. At this point I see him on ADS-B doing the same thing turning directly towards me. This whole time I cannot spot him visually as he is very close and likely above us in my blind spot. Again trying to avoid traffic I cut power to get lower and turn back onto base/final. He was following closely behind me the whole time even though he claimed to have me in sight on CTAF. I estimate that we was 500 behind me although I could not see him visually as he remained behind me on short final. I made a call on CTAF that I would be going around to avoid traffic then he gets on CTAF and says that he will also be going around. I then proceeded to communicate that if he was going around that I would be continuing to land full stop. When I once I rounded out I could actually see him out of my window less than 500 feet above us. At this point I am very frustrated with this guy so I decided to go back to ZZZ where at least I would have tower looking out for me. Back at ZZZ I heard him also making nonstandard radio calls to tower to the point they asked him if he was OK.