OK, fixed it!
Sorry for posting so trivial problem but it didn’t look like what it was.
It looks like it’s been caused by wrong settings. In Settings -> General, it was
WordPress Address (URL): http://www.uwolnijcialo.pl
Site Address (URL): uwolnijcialo.pl
After removing www. from the first one, all problems disappeared. Confirmed it by reverting to previous settings.
It also caused problems with login.
I hope it helps someone else 🙂