Oiling stops after a minute of milling

Hi all. I have a problem I cannot figure out myself so I need your help. Today I was milling a willow tree. After like a minute of cutting I sensed a smell of burning wood. Checked my chain and it was really hot and really dry and looked like burnt. I did the oiling test by running my saw with the tip of the bar facing a bright colored object and there were absolutely no signs of oil. I decided to clean my saw (the area around the sprocket, the oiler hole, the bar, etc.) and then did the oiling test again. This time it was oiling decently, I could see lots of oil flying off the tip of the bar. I started milling again and the same thing happened. After like a minute the chain got hot and there was no oil. There is a lot of oil all around the oiling hole, the part of the bar that is touching the saw body, but it seems that for some reason the oil cannot get onto the chain. What could be the problem and how to solve it? Is it because the saw gets clogged? But that doesn't make sense... It shouldn't get clogged so fast. Anyone has any tips?