Here is the sound I'm talking about.
I get this sound, unless I strum ridiculously lightly, on different amp volume settings, with no gain, but is seems to become quieter if I turn pick volume knobs below 50%. The worst part is that with gain it sounds completely horrible.
I don't remember when this started exactly, but I think I remember something similar happening on and off last year. Now it appears to be permanent.
I've already tried asking about this on reddit and got suggestions to try with another guitar and another cable.
Another guitar was not available, but I tried hooking it up to an electric keyboard. Here you can hear me playing it, while adjusting the keyboard's volume knob.
I hear some distortion starting on medium volume, but am not sure, if that's just because of incompatibility to guitar amp.
I've also tried hooking up the amp to a computer, using the 6.3mm guitar cable and a 3.5mm cable, and playing some sounds on the computer while tweaking volume on both devices. I've noticed no distortion during those tests.
The amp is a B.B. Blaster.
Based on the suggestions, I've tried playing in another room and with headphones. I could hear the distortion in both cases, headphones actually made it more audible. Additionally, I've checked the speakers and noticed a lot of dust behind the grills. I removed the dust, but the distortion didn't disappear. Didn't notice any other foreign objects.
Sometime next weekend I'm going to try and test the guitar and cable on another amp. Will post about the results.
So, I wasn't able to free some time last weekend, but I brought the guitar to a music shop yesterday. When the shop assistant plugged it in and started playing, there was none of the familiar distortion noticeable. I gave him the cable I was using to try with it. There was a noticeable signal loss, but still, no distortion.
I bought a new cable and tried playing with it after coming home. The distortion is still there. So, I guess the amp appears to be the most likely suspect now. I'll be bringing it to the store next week.
I brought both the guitar and the amp to the store. After plugging them together with a store cable I was able to reproduce the sound. What I understood from the assistants' explanation, the problem was with the amp's construction, it has got a peak volume, beyond which it can't go without the sound.