Flight instructor with student reported a near miss during landing at a non- towered airport with an aircraft landing on the opposite direction runway. The other aircraft went around to avoid a collision.

Narrative

I was in Aircraft X performing a flight for the purposes of training a student pilot. We were coming in to practice a landing at I23. We were inbound from the east; listened to the AWOS and CTAF frequencies to determine the active runway. Winds seemed to favor Runway 5 and no one was on frequency; so we chose to overfly the field and do a teardrop entry to the downwind for 5. We made several radio calls while overflying and entering the downwind and heard no calls for I23 on the CTAF but could hear other nearby airports that share the frequency. On downwind we saw another aircraft; Aircraft Y; flying over the downwind in the opposite direction at least 500ft higher than our altitude. Since they were not making calls; we assumed they were just overflying the field and continued our pattern to land. When crossing the threshold to touchdown our GPS gives a traffic advisory and looking at the opposite end of the runway as we are touching down; we see Aircraft Y on short final for the opposite runway. Since we are already on the ground; I begin to slam on the brakes to try to reduce the speed for a potential collision when the Aircraft Y pilot makes a radio call that they are going around. They sidestep the runway and fly over the taxiway next to us at no more than 300ft AGL. We make several radio calls to try to establish communication to the other pilot; all of which go unanswered. However; an additional aircraft soon makes calls on the CTAF and is able to hear us just fine. This could likely be an issue with the radio in Aircraft Y either not receiving or having the volume too low. A collision was avoided but could have been lower risk if communication had been a factor.