Do you have a controller connected by any chance? Unplug it and see if that solves it.
Thank you, this changed quite a lot. I did have a gamepad plugged in, and after unplugging it a lot has changed. The fps drop is not so sudden and inevitable now. I can play 2-3 matches qutie smoothly. However after that the performance issue strikes again. This time however it doesn't lead to such extreme overall fps drop. Instead my fps remain fairly high, unfortunately though, the frametimes go all over the place and the game becomes very jittery. That's one side of the problem and restarting the game somehow resets "the timer" so I can play few matches again. The other is that on few occasions, my fps went from 144, to around 50 and not only that, the frametimes went through the roof as well, introducing a very strong and constant stutter. To solve that I had to quickly go to the desktop and back to the game. This other issue may happen sooner than 3rd or 4th game.
Overall, this improved the situation quite a lot, however being able to paly only 2-3 matches without the issue (if I;m lucky not to encounter the other problem described) and then having to restart the game regularly is still an unacceptable state of affairs.