C172 Flight Instructor reported while on approach at a non-towered airport; non-communicating traffic departed opposite direction resulting in a NMAC. Flight Instructor took evasive action to avoid a collision.

Narrative

During a checkride preparation flight; I was executing the RNAV XX approach at ZZZ to minimums with the intent to go missed. Coordination with traffic on final (Aircraft Y) was established; and the pilot in Aircraft Y confirmed a full stop to allow us to safely complete the approach.At approximately 400 feet; Aircraft Z approached directly at our altitude with a separation of no more than two wingspans. This created an immediate and significant risk of mid-air collision.Aircraft Z made no communication; such as indicating they were holding short or departing. Had they done so; I would have promptly coordinated; communicated; and broken off the approach early to avoid conflict. Additionally; Aircraft Z made no attempt to avoid the potential collision and continued on the departure leg after Aircraft X broke off to the right in an effort to avoid contact.After the incident; I contacted the pilot of Aircraft Z to inquire if they were on frequency. The pilot stated; 'it was too busy for us to make calls.' When I asked him to repeat; he reiterated; 'it was too busy to be shooting approaches.' Further; Aircraft Y; which had coordinated with us for a full stop; reported being cut off by Aircraft Z. This could have forced Aircraft Y to go around; leaving no room for Aircraft X to safely maneuver.The actions (or lack thereof) by Aircraft Z demonstrated a failure in communication and situational awareness; and creating a hazardous environment for all aircraft involved.