A320 Captain reported during landing at night on a snow covered runway the aircraft did not stop until entering the overrun.

Narrative

ZZZ1 - ZZZ night landing in ZZZ during cold weather operations. I was reassigned to fly the ZZZ1 - ZZZ leg after my original flight to ZZZ2 was delayed several hours. I arrived at the aircraft approximately 37 minutes prior to push back and started downloading the paperwork. The First Officer (FO) had already loaded the FMC. Enroute; received report from Dispatch that ZZZ Runway Condition Code (RCC) was 5-5-3 for Runway 19. We confirmed by asking for airport conditions using ACARS. Based on our landing weight of 138.6 and and using the ZZZ 10-7D landing distance charts (no tailwind) we determined we needed a RCC of no less than 4 (this produced a landing distance around 5820 feet plus 220 feet for QNH correction; unless we burned fuel down to a landing weight less than 137.7 with no tailwind and QNH correction). Approximately 15 minutes later; Dispatch informed us that ZZZ had worked on runway and RCC was 5-5-4.We asked to be kept up to date on RCC if it changed and we would need to burn 900 lb of fuel if the RCC went back to 3. We also asked about using ZZZ3 (instead of ZZZ4) as an alternate if we needed to lower our landing weight more. In the end; we kept ZZZ4 as an alternate since we would be able to burn fuel down to an acceptable landing weight and still land in ZZZ4 with 4500 - 5000 lb of fuel with a BINGO of 8000 lb of fuel. We elected to descend early to burn more fuel and lower landing weight to around 138.3. We performed the ILS Z Runway XX. Broke out of overcast between ZZZZZ and ZZZZZ1 with moderate snow around 2500 feet AGL. FO asked for another braking action report and Tower said it was still 5-5-4. Autopilot and auto thrust off; on airspeed and glide slope with aircraft getting slightly high around 1100 feet; then back on glide slope by 900 feet. At 200 feet; I shifted the aim point to 500 feet from the approach end. At this point the FO asked Tower to lower the runway light intensity because it was hard to make out details on the snow covered runway with the bright lights. I noticed the runway's center 2000 feet was partially cleared of snow; but the approach and departure ends of the runway (~2000 feet on each end) were snow covered. Never noticed the light intensity decreasing. Landed approximately 1300 feet down runway (hard to tell exactly with snow covered runway) and close to VLS. Medium autobrakes were selected as per the 10-7 page; but I did not feel a deceleration or see the decel light illuminate so I immediately applied manual brakes to start the deceleration.Brakes seemed to work normally entering the center 2000 feet of the runway; but with 3000 feet remaining; the FO and I both noticed the brake's effectiveness was not what we needed to stop the aircraft on the runway. At this point we both applied full manually braking and the anti-skid was cycling. I said we are going into the overrun about 500 feet from end of runway. I continued using full reverse thrust and maximum manual braking. The aircraft entered the overrun around 15 - 20 knots until we stopped approximately 200 feet onto the paved overrun. At this point I moved the thrust levers to forward idle. We informed the Tower that we had stopped in the overrun and we were instructed to turn around and taxi to the gate. After informing Tower of the RCC being less than 4; we said the RCC was 1 or 2; the Tower noted we were fast on the runway (we were; due to the poor braking conditions). We performed a 360 degree turn in the overrun and taxied uneventfully to the gate. I performed walk-around after engine shutdown and no aircraft damage was observed.