How to get a flat-head nail out of a piece of wood?
This style of nail puller does a good job of pulling out nails that are flush or deeper.

It won't leave the original surface completely untouched, as it needs a tiny bit of clearance to get a grip on the nail.
And here are a few videos demonstrating how to use it.

https://youtu.be/sWC0hkTH250
https://youtu.be/573UbJ3tgQI
https://youtu.be/kPh8GkComL0