Long and short of it............bad lower end of the 4.6 Triton. Your oil pump and crank bearings are the only items that are common in oil pressure loss. This motor has no OHV style camshaft, so all the oil pressure is created in the pump and bearings. (Modified and some Windsors would turn the cam bearings, causing the loss of oil pressure, but this motor doesn't have that possibility.
You could check the oil pump for grooves cut into the housing. Most likely you will find this to be your problem. Problem is that the bearings have run with low or no pressure, so they are most likely worn to the copper also. Any excessive wear and you will lose all you pressure when the motor comes up to full operating temperature or long run.
I hate to say it, but looks like you need a short block at the least. You may be able to cobb it together with another pump, but the wear on the bearings won't allow the pump to keep up over time.
One more thing, use a Motorcraft oil filter when you change the oil. These have a back flow check valve, so the oil pressure is immediate when started cold, and not looking for oil pressure, like with a Fram or Wix or Napa. There is a forum on filters on this site, that states that those brands are the worst for Ford motors. Purolator was rated good, as was Mann & Amsoil, but not the K & N's or the AC-Delco OIL filters. The air filters were not listed in the tests.
Hope that helps you.