General aviation flight instructor reported a NMAC during landing at a non-towered airport with an aircraft that performed a go-around behind them. Both aircraft took evasive action.

Narrative

I was on a training flight in a Cirrus SR20 staying in the traffic pattern at ZZZ. My student and I had previously made 6 or more trips in the pattern; mostly touch and goes; and always reporting our position. There was another C172 in the pattern; and they were also making position reports and the separation was excellent. There was a Bonanza" who reported 15 miles to the north west and that they would continue inbound; they never stated a runway or traffic pattern entry. We continued in the pattern; as we expected them to be a few minutes and to overfly the field. They made a 10 mile call out saying there were going to make traffic for [Runway] XX; which does not exist at that airport. I would have asked for clarification; but my student was actively landing the airplane. Upon doing another trip in the pattern; and reporting abeam; base; and final; we hear Bonanza asked for a position report from us. My student was flaring the airplane; and I told her to keep flying. After we made our touch and go like we previously said; we hear that Bonanza was making 'right go around'. After I assure that the aircraft is stable and the student has control; I look to our five o'clock and see the traffic only a couple hundred feet behind us. It was close enough to see the pilot in the front seat; and the paint color. I made clear callout that we would continue to climb on the present heading; and then depart the pattern to the north upon reaching 2000. The traffic reported that they would do the same. However; when the other traffic reached 1700; they turned left to continue in the pattern."