As you state in the question, it is the low oil PRESSURE warning, not low oil level.
Although low oil level can cause low pressure through starvation, this is clearly not the case here.
There are a number of possibilities.
When was the last oil and filter change? If the oil is old, of the wrong grade or of poor quality you will have this problem.
If the engine is worn, especially on the main bearings, you will have this problem, but it's usually only seen on badly maintained and high mileage vehicles, or on vehicles which have been driven with low oil pressure (it's a vicious circle).
Failing oil pump.
Faulty oil pressure switch.
Blocked oil strainer in sump, either by foreign matter inside sump or because sump pan has been pushed in, caused by hitting something or incompetently placed mechanics jack.
epsilon
September 2009