Hi Herman,
That is really strange. However, your Nano is not lost -just buy a cheap power bank that puts out 5v and use that to power the Nano, which really runs from 5v, not 3.7v.
In fact, you can always buy a very inexpensive and small charger-inverter board from the likes of banggood for around $2.
Yes, I know it's a pain in the butt but at least you can keep using your unit.
These boards use the 54xx chip..
Note both boards use a 4R7 coil but the cheaper board uses a 33ohm in series with USB 5v but the more expensive board uses 2 - 1ohm in parallel for current limiting.
I wonder if these chips are sensitive to input current from USB and are being damaged somehow.
Try placing a 33ohm at the 5v input to the chip and see if that helps.
You might be able to transplant parts from one of these modules to the Nano.
Regards
Larry