A Flight Instructor reported an NMAC event during initial climb with opposite direction landing aircraft. Flight Instructor stated the landing aircraft provided inaccurate position radio calls.

Narrative

I was doing a training flight with my student. The airport we operate out is uncontrolled (XXX). The winds were very light and favoring Runway XX. Me and my student planned on taking off Runway XX. We transmit our Intentions on CTAF that we are planning taking off Runway XX. I hear the other aircraft on the radio and they say plan on landing Runway XY. Since they were coming from the north they announced that they plan over flying the field and entering the downwind. So I thought that they are probably going to be another 15 minutes give or take and not be factor. So I go ahead and do another radio call that we are departing Runway XX. So on the upwind portion of the pattern at around 500 feet AGL me and my student see the aircraft on the final for Runway XY. I Immediately take controls continue my climb and turn to the right. After investigating the incident I have discovered that the aircraft decided to enter right base for XY and land. The other aircraft did not go around they continued their approach and landed.