General aviation pilot reported a near miss on final approach at a non-tower airport while proficiency training. The pilot performed an evasive maneuver at low altitude to avoid the other aircraft; re-entered the pattern and landed safely.

Narrative

While practicing VFR ILS approaches into MCC (no view limiting of any kind); on short final; another aircraft turns final directly onto me; and I am forced to take extreme action (45 degrees or more of bank) to avoid a collision; while only ~500ft AGL; with full flaps. In retrospect this was an extremely dangerous situation; and I am very confident had I not taken any steps to avoid this collision; there is a high likelihood an accident would have occurred.From memory; I made 10 mile; 5 mile; 3 mile; and short final callouts on the CTAF 122.975. I do not believe there was any radio issue on my end; as I had just spoken to someone else on CTAF shortly beforehand (an instructor I know flying another airplane).Someone calls their base; I again inform them I am on short final; my flight logs show me about 0.5nm from the numbers; seemingly ignoring my communications; they call their final. I get a TCAS alert from Foreflight (connected directly to my plane's GPS & ADS-B receiver via a GTX 345 + FlightStream); confirm it visually; and see them on a rapid collision course for where I was going to be in a few seconds; and did a hard 360 to avoid.At about XA:46Z time; limited ADS-B replay data shows me at 600ft and Aircraft Y; the plane in question; at 1125ft. Upon looking up the tail number; it seems Aircraft Y is a low wing. It is very well possible that the pilot did not see me due to this fact; as forward-below visibility can be reduced for certain low wings. It is also possible he either did not receive my communications; or did not heed them appropriately. In retrospect; there are a few things I could've done better: Although I made an attempt by trying to call my short final multiple times; I could've communicated to the pilot on base directly that if he wanted to avoid a collision; he should hold off on turning base until he had me in sight as passing his wing. I could've retracted my flaps BEFORE taking corrective action. I could have taken corrective action as soon as he called base; albeit there was not much time between the base and final call; so I was still processing the decision the other pilot made-- was he trying to get there first; or did he just not know I was there? I could have broken off to the left instead of the right in this particular case. After the collision was avoided; I should have immediately discussed it with the pilot in question. I was a bit shocked; and due to adrenaline dump; I instead opted to rejoin the traffic pattern on a 45 and immediately land so I could collect my thoughts.