2006 Toyota Rav 4 burning oil question

Hi I have a 2006 Toyota Rav 4. At around 94,000km I noticed the oil pressure warning light coming on and off when I step on the brake. It's not on all the time. When I took it to the Toyota dealer they said the engine has no oil. No leaks whatsoever. They said I over run it by 4,000km from the recommended change oil interval.

Thus my first question, is that true? That if you run it longer than the recommended mileage it will ran out of oil? I've had corollas and camry before and even if I overrun it by 5,000kms I've never had one that "lost" the oil in the engine. It was dark and dirty but there is still oil in it. By the way I live in Ontario, Canada.

They said at that time, no engine damage was noted. 4,000+ kms after I checked my oil and it was below the "low" limit. Is it now burning the oil? If yes, will that cost a lot to fix? Or can I just add oil every 3,000 kms, if that's the case will the car last for another 3-5 years?