There is nothing about a five-string P-bass pickup specifically that should contribute to hum. Note: many humbucking soapbars designed for five string basses are similar in that they are side-by-side split coils, one covering three strings and the other covering two.
As noted by others, have the wiring checked. Also... avoid touching the poles.
That's a downside to exposed-pole pickups. It is weird that it happens to one coil but not the other, though.