Arduino Nano and SIM800L on Battery Power
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  Thin battery to power arduino micro with nRF8001?


Also, I found a 6v battery (A544) at Menards that is about the same diameter as a AA, but is considerably shorter. It has 178 mAh capacity.

I haven't taken the time to test it, but I would guess it wouldn't have as much of a temperature issue.

It's not as small as the CR2032 in the above mentioned link, nor is it flat, but it's an option.

You might consider an Arduino Beetle. They are really super incredibly tiny and still 16mHz. I can't get any more info from where I'm at, but I'd believe they would use even less power than a Nano/Micro.

I found an Instructable that shows how to use a Beetle, a rechargeable 3.7v LiPo battery, and an HC-06 Bluetooth module. Sounds like it's equivalent to what you're looking for.

http://www.instructables.com/id/Beetle-Minimize-Your-Arduino-Projects/