Nothing jumps out at me. Double check polarity and measure the actual voltage of both batteries if you can.
Do you have anything else connected other than the battery and Nano?
Measure both Vin and the regulated supply (3.3V and/or 5V) with a voltmeter in both cases. This will give us a much better idea of what is going on. If you don't have a volt meter, well it's time to buy one.
Nothing else connected. I have a multimeter, and the results are:
a) With Alkaline, 5vpin = 5.06v, 3v3pin = 3.31v
b) With LiPo, 5vpin = 0.925v, 3v3pin = 0.915v
Today I've received my LiPo fully charged (now it provides 12.3V). Also I've mounted a simply resistor circuit with 216 ohms, and it provides about 56mA, which seems correct.
After that, I've put again the LiPo in the Arduino Nano Vin & GND pins, and still failing. I only see the pwr led light for a fraction of second, then nothing.
Again, I test a 9V battery in the same pins, no problem!
I don't understand anything. Why I cannot get the LiPo battery to power the circuit?????
If I put a serial resistor of 100 ohms, then everything goes ok!
I'm starting to believe that the circuit has a limitation of about 10V or more than 50mA in the Vin Pin.
I'm no battery expert, but it may have to do with your mAh rating? A "typical" 9V s rated between 565mAh (Alkaline) to 1200 mAh (Lithium). Your LiPo is a mere 180. In a perfect world, you should be able to run two hours on that (180/76). But reality is never that simple....especially when drawing such a high percentage.
I don't think it is the mAh rating. As I say, adding a serial resistor makes the difference!
Without a serial resistor there were 75mA and 12,2V in the Vin pin. Adding a 100 ohms resistor limits the current to 46mA and 7.52V.
So I believe there must be either a limitation on voltage or current in the Vin pin, though I'm not certain of that.
PS: in the Arduino nano specs I see that there is a DC Current per I/O Pin of 40 mA. I don't know if this also affects to the Vin pin.
Read with your meter, the voltage across the battery, when you hook it to the nano. Does the battery voltage stay high while hooked to the nano?
May want to do this on both batteries for a comparison.
Specified the maximum you can drive (or sink) from an output, without [type] causing a problem with the IC.
You say nothing is hooked to your Nano, so there will be 0ma on the pins.
By himself the ciruit works, as with 9V LiPo the program inside runs smoothly (included blink program).
About voltage checks:
9V battery
1.a) Alone = 8.37V
1.b) With Nano - without serial res 100 ohms = 8.19V
1.c) With Nano - with serial res 100 ohms = 8.07V
LiPo battery
2.a) Alone = 12.25V
2.b) With Nano - without serial res 100 ohms = 12.21V
2.c) With Nano - with serial res 100 ohms = 12.19V
And now the weird things:
I was testing case 2.b (the one I wanted), and the circuit started to work intermittently! Without touching anything. I've double checked loose ends, none. The same circuit, with a serial resistor, runs smoothly.