Is my oil pump failing

Hi folks! I feel like I already know the answer to this but I also feel like I could be an idiot.
So, I lived in the Benz for a few months. I didn’t drive it but I did run it to charge the battery and use the heater.
Along the way, the oil light started coming on. It was long overdue for a change. It also started “smoking”. White colored exhaust. The oil light would turn off after about 5 minutes.
Anyway, fast forward, changed the oil, got employed and don’t live in it. Now I drive it fairly regularly.
After a couple of weeks of regular driving the smoking stopped entirely. It was always intermittent and only ever happened after the car was warmed up. The oil level is correct and the car runs and sounds great. No water in oil or oil in water. But the oil light still comes on after about 10 minutes of driving. Then goes off about 5 minutes later and doesn’t come on again.
I think it’s the oil sensor and I know it’s a cheap and easy thing to eliminate as the cause but for reasons that would take too long to explain I’d like other opinions. Thanks in advance for your time and help!

If you actually has low oil pressure, it would not go away after 5 more minutes of driving.

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Replace the sensor and see what happens.
If possible, have a mechanic hook up an oil pressure gauge.
As far as the white smoke, just a guess, condensation in the exhaust system from sitting.

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And he wants rational opinions are saying that?

Actually one morning I had the oil light come on in my 86 riviera. It was new enough that I swung by the dealer before driving to work. They just told me it was the sensor and don’t worry about it. I replaced the sending unit and all was fine.

I bet OP’s wife told him to change oil pressure sensor.

My Corolla has been laid up, not driven for 4 years now due to Covid related complications. Not living in it, but I’ve been idling the engine for 20 minutes once a week to keep everything lubed and percolating. After about 18 months of this, I started to notice excess tailpipe smoke during the idlings. I traced this down to what I believed to be condensation forming in the exhaust system from the long idles, made worse by parking the car on a slope, with the tailpipe higher than the engine. This prevented the exhaust system from draining properly. I turned the car around, so the tailpipe was lower than the engine, problem solved, no more smoke from the tailpipe, after just a couple more idles. I’m thinking your exhaust smoke problem was the same cause, and now you are driving it, solved.

As far as the oil pressure warning light, that’s something that must be addressed asap. You could either ask a shop to measure the oil pressure using their own shop gauge. Or just replace the sensor and see what happens. One other idea, S=sometimes the oil pressure sensor’s rubber diaphragm breaks, and the sensor becomes inaccurate & also leaks oil. If you can locate the sensor, see if it is the source of leaking oil.

signs of oil pump failier is low oil pressure light or noise from the valve train and engine tempreture