Piper Arrow Pilot with Instructor reported an alternator failure during flight. After determining only battery power was available; the decision was made to return to the departure airport where light gun signals were required due to loss of comms.

Narrative

I went on a complex airplane training lesson with my instructor. Earlier in the day; the manger texted us and told us that the plane was in Maintenance and we would be the first flight after Maintenance was complete. When we got to the flight line; I performed preflight and run up; systems appeared normal. The Garmin G5 configured as DG was INOP and known to Maintenance; not required for our day VFR flight. However the amp gauge was at zero; we saw a small tick up when we clicked the pitot heat on and took that as; the gauge is reading and detecting positive charge.I took off and flew to the practice area without incident; I performed slow flight; power off; power on stalls; and steep turns. I turned back towards the airport ~15-20 miles; and planned to perform 3 touch and goes. On the way back the Garmin G5 AI threw an exclamation mark warning but I were in the middle of cleaning up the airplane and disregarded it thinking it was a message alert from the 650 GPS to switch tanks. Several minutes after; the Garmin 650 GPS threw several failed systems warnings; the transponder failed; and the instructor side comms failed. The student side comms was working but intermittent. While I was flying towards the airport; my instructor attempted to power cycle the masters; avionics; & turn off the lights to reduce the load. It helped slightly and we were able to contact ZZZ Tower and tell them we were having electrical malfunction and heading towards the airport. We attempted to squat 7600 but the transponder died as we were setting it up. My instructor took the flight controls and I attempted to contact Tower. They cleared us XXL and we had several garbled transmissions to Tower afterwards. Eventually; the G5 AI powered down; we confirmed the alternator was not charging and the aircraft was running on battery only. We shortly after had a battery failure and total loss of electrical. At a 5 mile final; we attempted to lower the gear and realized that we would not get gear down indication. We attempted to contact the Tower for a low pass but was not able to confirm they received the message since the comms went out. We set up for a 1 mi final on XXL; IFR runway; received the light gun green indication; but saw a jet was setup for takeoff on XXL and sidestepped to XXR. My instructor performed a low pass; and entered the pattern for XXR. I found the emergency checklist and we verified that we performed the loss of power gear down checklist; Gear position down; actuate the emergency gear handle. On the second pass to XXR; my instructor elected to land; no flaps. We received the light gun green and processed to land. There was fire trucks waiting for our arrival. We taxied with flashing green light gun and parked without further incident. What I could do better:Aviate; navigate; communicate. The startle of the failure sent us into troubleshoot mode right away. I started running through troubleshooting right away but didn't take a moment to asses the situation and work checklists.It took me; a long time to pull the checklist and start working through items. Poor checklist discipline was there but I eventually found what I needed. Poor CRM; there were several times we both went heads down; clear pilot flying; pilot troubleshooting would have been better.Better systems knowledge; I often approach these plane; and gauges with skepticism if they are really working or not. The zero amp reading could have been a real error we discounted or leaned into confirmation bias to continue the flight. There was also an indication of low power on the G5 Attitude indicator but I discounted it as I thought it was a message indication from the regular tank switch messages.What others could do better:Maintenance: no post maintenance flight was performed. After we landed we learned the nature of the repair was electrical but there must have been something missed.Tower: The runway and approach was not cleared for us on entry. We crossed theactive IFR jet route to the smaller training runway. A jet was cleared for takeoff on the runway we were approaching even with the green light gun. On our second approach; they continued IFR arrivals while we were NORDO in the pattern.