What is the thickness of an Arduino Nano board?

That’s an unusual question…

I don’t have a genuine Arduino Nano, but I’m sure the dimension would be similar to this clone.

If you get them without the header pins preordered the thinnest Arduino Nano would be 7.05mm (including component heights on the underside of the board)

If minimal height is what you’re after you could get something like a Pro Mini that does not come with USB port. You will need a USB to serial board to program it though.