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Just for others coming here in the future: I changed the connections as suggested by @mfalkvidd , restarted the node and now the radio runs from batteries and the Arduino is powered from USB (phone charger).

Unfortunately this doesn't seem to have improved the radio communication, but this is another story...
Stefano

@mfalkvidd no, at the moment it's connected to the negative pole of the battery.
So the V+ of the radio to to the positive of the battery and the GND to the negativa