Problem could be a bad oil pressure sensor or the oil pump is failing. A mechanical oil pressure gauge needs to be hooked up to verify actual oil pressure. Problem could also be a bad stepper motor in the instrument cluster that drives the gauge.
Can you see the smoke colored sensor near the windshield? I've had one break and fall down which caused this exact problem. If you can see it, pull the plastic trim piece off, make sure the connections are actually connected, have seen these come unhooked.