Hola guys! Thanks for your replies. I've tried today to use better power source and slighly higher voltage and much bigger cap.
I've used my soundcard power supply (9V, 4A) and MP1584-based chinese step down regulator and 2.2mF capacitor. Voltage was 3.33V.
And behavior was absolutely the same.
... and, while writing this post I've realized, that I have significant voltage drop on these "arduino jumper wires". 3.3V on regulator output, 3.1 and even lower at end of 10cm (+- 4in)!
So, I shortened the wires to extreme short jumpers... and now, IT WORKS!