So I decided to inspect the roots and repot it.
Something bad happened to the roots - they all came off easily when I touched them - they were all dead. However the good news is the cactus recovered itself, there are a lot of new roots coming out of its bottom (root base?) and from the sides. What I noticed on the old roots are tiny clusters of some white like.. hairs or moss, I think it's some kind of fungus or root mealybugs (I really have no experience to assess). However it's all gone with the old roots.
Now I'm pretty sure it's my fault. This cactus was probably overwatered (although I never water more often than every two/three weeks) but the thing is - the soil was quite bad compared to other cacti I planted later, too organic thus keeping water for too long. I think that might be the reason and that's a good lesson for me. Seeing the new roots, I'm really hopeful that it will recover and I'm quite emotional towards it since it's one of the first cacti in my small collection
I'm going to leave it for a few days, then pot it and leave it for 1 or 2 weeks without water and then slowly start watering.
Thank you guys a lot for help and if you have any suggestions about what to do next, please share.