Hello Tom
The voltage levels were correct . In another post I read that toggling RESET pin before sending configuration is desired (at least 50ns up) - that is what I did. The STBY pin (as I wrote in the attached file) was connected to +3,3V (BVDD).
Finally I was so frustrated that I decided to change ADC converter to ADS 8568. It is much better described (also on the TI forum), so I spend little time to connect it, send configuration and make it work. I also did not connect all pins (CH_xy, REFzP).
Anyway I still do not know what I did wrong trying to make the ADS8556 work. I found another topic about the ADC where this issue (how to set control register) was still unsolved.
http://e2e.ti.com/support/data_converters/precision_data_converters/f/73/t/189329.aspx?pi73718=1
If anyone find similar issue and solution, please share with us in this or in the aforementioned topic.
Thank you very much for your efforts!
Mariusz