GA Flight Instructor with student reported an NMAC during departure climb from JCA non-towered airport requiring evasive action. Reporter stated language barrier was a factor in the incident.

Narrative

Pattern work with my student at JCA. 4 planes in in the pattern; two flight school planes and 2 pilots who do not speak English well. Final lap in the pattern for Runway 35; We announce a left crosswind departure to the southwest. We begin our left crosswind turn to the west ; and I hear 'Departure leg Runway 35 Aircraft Y.' I look back to find the traffic; and see Aircraft Y. I then hear 'Jackson county left crosswind 35 Aircraft Y.' Knowing he is in a twin; I know he is moving and climbing faster than me. I Then announce ' Jackson traffic Aircraft X is in the left crosswind departure for 35 to the west; do you see us?' 'No;' is the response my student and I get back on the CTAF. I see no corrective action on the ADS-B traffic map on our MFD (Multi-function Flight Display). I announce 'You are climbing into us.' no response. Looking at the onboard ADS-B traffic map I see no corrective action; and we are within or close to TPA of 2000 MSL; Aircraft Y kept ascending until he caught up to us. I command 'my controls' from my student; and pulled a maneuver to gain as much altitude as possible; knowing he was remaining in the pattern. I ultimately read '00' on the altitude difference; and the the plane on the MFD was directly atop ours. I could not see him behind me. I can only assume it was a miss of 400 feet. I truly have no idea how close they were. 400 feet of a near miss feels generous. I then hear 'left downwind 35 Jackson.' At this point I am close to 2300; I believe; and finally locate Aircraft Y. I then ask 'do you see me off your right wing?' 'I do now.' Was the response. I do not know if a lack of English proficiency; blatant disregard for listening to other planes in the pattern; or poor ADM (Aeronautical Decision Making) contributed. Potentially all three. No decision to extend the upwind. I can only climb so fast in a single engine. Left is the downwind where I know Aircraft Y wants to go. Right is the corrective action he would make if any would come from him. Descending puts me on top of his airplane. If this was a plane with no radios; I can only do so much. however; it was established that this plane has radios; he can make the calls; and he can hear me. From my understanding this airplane had a spacing issue coming back into ZZZ as well.