I tend to agree with Snarky that there must be a Schottky diode reverse current protection beyond the barrel jack (I suspect D32 on the front of the carrier board is the culprit). I cannot see power regulators near that section of the board so it would seem that the system required regulated 5V input (but clearly we need at least about 5.3 as my experience shows).
So straight 5V will probably not work for you. 6V would be too high. But with 6V you can put a silicon diode in line with the + side of the power supply (anode on the supply side and cathode on the nano side) and that will cut it to about the right voltage you need. Beware however you will need a fairly hefty diode that can dissipate about 3 W (probably not the usual small axial diode, to be safe, one with a heat sink – and make sure it is a normal silicon diode and not a Schottky diode).
Nevertheless if you have a 11-12V battery you should be able to use a step-down (Buck) converter. For example https://www.ebay.com/itm/Adjustable-DC-DC-10A-7V-32V-Converter-Buck-Step-Down-Regulator-Power-Module-XODF/122850143075?hash=item1c9a706b63:g:E7wAAOSwjodaKjma
Note this will not work with 5V or 6V batteries --need at least 7-9V here
You can always buy pre-made plug wires to screw into the terminals and feed the barrel jack
For USB: these can be handy
https://www.ebay.com/itm/10pcs-Black-70cm-USB-2-0-A-Type-Male-to-4-Wire-4-Pin-DIY-Power-Cable-Cord-28AWG/323307308915?epid=1476974584&hash=item4b469e0773:g:ll0AAOSwanRXgh-~