What do you mean by "barely any"? Was there still oil showing on the dipstick? If so, then you weren't more than about one quart low, which isn't enough to trigger the low pressure warning.

 

The stick does not go to the bottom of the sump, it just needs to "dip" (get it?) into the very top layer to get a reading.

The low on the stick means, hey, you're a bit low, might want to check on things. The idiot light is, "WHOA, there is a major issue, turn me off now!"

 

The low pressure warning will only come on when the oil level is so low that the pump starts to suck air along with/instead of oil. Until then, you will have full pressure.

I hope you're not making the mistake so many do of equating the oil pressure warning system with a low oil level warning system. They are two, entirely different things and most cars do NOT have an oil level warning system. On the Mini, and most other cars, you can actually let the oil level fall below the bottom mark on the dipstick and have plenty of oil pressure. However, it does mean that you're approaching the oil level point where eventually, the lack of sufficient quantity will result in a precipitous loss of pressure. ANY time you see the oil pressure warning light come on, pull over to the side of the road as fast as is safe to do so and turn the engine off IMMEDIATELY. Do NOT continue driving for any reason than an outright emergency. Otherwise, permanent and very expensive engine damage will result.

My wife's best friend was ignorant of the true meaning of the oil warning light on her nice Volvo. (She loved that car.) When the warning light came on (due to lack of maintenance), she thought it simply meant that the oil level was getting low and she should check it fairly soon. She continued to drive. As a result, she completely destroyed her engine and because replacing the engine was going to be soooooo expensive, she sold the car for a huge loss and bought a cheap used economy car because she could no longer afford anything else. (The huge loss on the Volvo.)

Make sure you and everyone you know is aware that the oil warning light 99 percent of cars is NOT a level warning. It is a pressure warning, and like your body and it's blood circulatory system, no pressure means impending and rapid death to engine unless you shut if off immediately.