Light aircraft Instructor pilot reported a NMAC with another light aircraft in the pattern at VPC airport.

Narrative

Aircraft X on student pilot training flight with CFI doing closed traffic pattern work. Aircraft X was on left downwind. We reported turning crosswind and then reported turning downwind. At midfield downwind we heard Aircraft Y report entering the downwind leg. We reported our midfield downwind position via CTAF (122.7). About 10 seconds later the Aircraft Y flew over the top of [us] by approximately 100'. Three calls were made on the CTAF asking the pilot of Aircraft Y if he/she was aware they just overflew. There was no acknowledgment from the Aircraft Y pilot. After the third call; Aircraft Y reported they were breaking off short final and would depart from their final approach to the SE. After making about a 20 degree turn to the right they returned to their final leg for runway 1 and executed a touch & go.After landing; [I] called Unicom for a radio check who responded with; 'Hear you loud and clear.'Aircraft X does not have an engine driven electrical system and does not have a transponder or ADSB-out. A battery supports the flashing beacon atop the fuselage and a panel mounted handheld VHF transceiver.