Flight Instructor reported a near miss in the traffic pattern at P08 non-towered airport when another aircraft turned in front of them. Instructor took evasive action to avoid a collision and departed the pattern.

Narrative

While conducting practice takeoffs and landings at XA:58; at P08; (Coolidge airport); Aircraft X had an apparent near-miss with Aircraft Y.This was due to a combination of factors; as listed below:The scenario began at approximately XA:56. Aircraft X had taken off from Runway 23 and was on the departure end of Runway 23 after a touch-and-go landing.Aircraft X was unable to extend the departure leg due to Aircraft Z attempting a straight-in practice instrument low-approach to Runway 23. PIC made the determination that extending the departure leg would pose a conflict.Aircraft X initiated a crosswind turn at XA:57; with Aircraft Z following behind them. At this time; Aircraft Y entered the pattern; disregarding guidance set forth by AC 90-66 (10.6) and FAR 91.113.Aircraft X entered the right downwind for Runway 23; and Aircraft Y terminated their practice instrument approach and turned left at XA:58; climbing directly toward Aircraft X.Aircraft X was unable to do the following:- Could not continue straight due to Aircraft A entering the pattern ahead of them. - Could not complete a left 360-degree turn for spacing due to Aircraft Z flying behind them. - Could not turn right to overfly the pattern due to parachute operations being conducted at TPA (Traffic Pattern Altitude). At this time Aircraft X made a right turn and began climbing to create space between them and departing traffic. Once Aircraft Y had cleared the pattern; Aircraft X continued back on the downwind; but due to Aircraft A and another aircraft inbound on a practice approach; opted to depart the pattern.