This can only be one of three possibilities:
1) Bad oil pressure sensor
2) short in wiring to oil pressure sensor
3) low oil pressure.
The easiest way to find out if it's a bad sensor is to remove the sensor and attach a pressure gauge to the port. Then you'll know for sure. If the pressure is low, you'll have to have an oil leakdown test performed to determine the condition of the bearings. If they're worn, a new oil pump won't solve your problem.
Have you been good about oil changes? If not, you may be paying the price now.