2001 Ford Mustang no oil pressure

2001 Ford Mustang mileage: 38,500.I was driving on the highway when my oil pressure dropped to zero and I heard a tapping so I pulled over checked the oil and it read full.I waited a little while and went further when I noticed white smoke coming from the exhaust. I put a quart of oil in and it made the noise stop but my pressure still was at zero. At first I thought it was just the gauge but then the white smoke was thick and the tapping started again, I pulled off the highway and left the car. It is still registering full.I thought it was a head gasket. My dad thinks it is the oil pump. Have blown a rod before and I do not think that is the problem. Do you believe it could be the head gasket or is it more likely the oil pump and will I have to replace the engine?Thank you!