GA pilot reported while on final taking evasive action to avoid conflicting NORDO traffic turning base to final resulted NMAC.

Narrative

I was VFR enroute from ZZZ1 to ZZZ. After departing ZZZ1 I contacted ZZZ Approach for traffic advisory on any commercial traffic enroute to ZZZ. I was assigned a discrete transponder code and handed off to ZZZ2 Tower for a midfield crossing south-north over ZZZ2. After crossing ZZZ2 midfield; ZZZ2 Tower advised me to contact ZZZ advisory frequency. I made my first call on the advisory frequency and indicated I was 1 NM east of ZZZ and entering left downwind for Runway X. After passing Runway X threshold on downwind another aircraft announced they were 'setting up for a left base'. I made visual contact with the aircraft; which was on an extended left base for Runway X (approximately 2 miles west of the threshold of Runway X). I announced I would extend my downwind for spacing for the aircraft on left base.I announced and turned left base after the previous aircraft was ~1/4 NM from the threshold of Runway X. I then announced 3 NM final for Runway X. Within a few seconds of that call; I heard another aircraft announce left base for Runway X. As I was descending on final I visually identified another aircraft turning final for Runway X off my port side at my altitude and a few plane lengths ahead. I called out to the aircraft turning final and advised I was already on final and suggested they break off their approach. There was no reply from the other aircraft. The other aircraft continued approaching my port side and when they were within ~100 ft. laterally I announced a balked landing. I was crossing over the freeway at approximately 200 ft. MSL at the time. I applied power; made a climbing 180-degree right turn over the river and climbed to approximately 400 ft. MSL while attempting to stay within the SFRA and outside the Class C airspace over the river. I listened and visually verified no traffic in the pattern on left base for Runway X announced a right base for Runway X and landed without further incident. While this was my first time landing at ZZZ I had briefed FAR Part 93 for air traffic rules and communications within the SFRA prior to departing ZZZ1 and was in contact with ZZZ ATC en route. After searching FlightAware it appears as if the other aircraft was performing a low approach or touch and go at ZZZ just after I had entered left downwind for Runway X. I do not know if they heard my call for extended downwind for Runway X but my best guess is that they did not; as they turned base while I was on extended final and turned final into my flight path.