Pilot instructor reported an NMAC at 1RB non-towered airport.

Narrative

Instructing Pilot (IP) was conducting an instrument rating flight lesson with Student Pilot (SP) in Aircraft X. SP was flying practice approach RNAV 26 at 1R8; circle to land same runway; as published starting from IAF RERME. IP was making appropriate traffic pattern calls on CTAF frequency 122.8; including a 10 mi; 5 mi; 2 mi; and left downwind entry call. No other traffic at 1R8 was visible on ADS-B or making calls on CTAF through the duration of the approach. After maneuvering into the downwind and preparing for a base turn; IP declared that crew was beginning an early left base for runway 26 on CTAF before the student initiated the turn from downwind to final. Company traffic was holding at RERME and actively communicating with IP to coordinate flow of approaches. Aircraft Y did not make a position call but stated that they were 'looking for [us].' After completing the left turn to the base leg; Aircraft Y was seen turning to a right base from a very tight downwind leg. IP tried calling to Aircraft Y on CTAF to make them aware that we were right in front of them. Aircraft Y stated that he would continue. With no room to maneuver; SP and IP decided to continue for the runway and turn to final early as we would not have been able to out-climb Aircraft Y. IP stated that aircraft was not going around as there was no room. Upon looking through back window; IP could see Aircraft Y no more than 100 feet away from aircraft. Aircraft Y seemed fully aware of our presence but was going to continue to land. When SP/IP were wheels down on the runway; IP could still see other aircraft - aircraft was so close; IP could see facial features of the pilot. Aircraft Y was estimated to be no more than 100 feet away based on runway center line markings. After IP called out the pilot over CTAF one final time saying that there was no room on the runway; Aircraft Y side-stepped runway and continued low over the grass to the south of the runway. As SP/IP were rolling down the runway; Aircraft Y could be seen performing a non-standard maneuver to turn back to land on Runway 8; against the normal flow of traffic and without regard to wind direction/velocity.