Where is the problem - the guage or oil pump ?

Well the oil pump is just a set of gears that force the oil through...seldom fail in this manner...in other words doesn't start to go bad then get better, go from bad to not so good, but never will have zero PSI oil pressure unless at idle speed, even then would have SOME pressure...so it's your oil pressure gauge at fault. Check ground connection to chassis, from the battery...follow the black wire until it connects to the chassis. undo wirebrush and re-assemble for solid chassis ground...hopefully this half of the puzzle will solve your problem~

While you're at it the lug which connects to the engine block should be wirebrushed clean as well----~