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I have a 1/4” (~6 mm) nail that someone put in a wooden bed frame that is 1/2” (12 mm) thick.
The wood is hard (not soft) and the nail is flush with the wood in some parts, actually digging into the wood in other parts.

How does one get this nail out without damaging the wood too much?

The only thing I can think of is to “chisel” around the nail head until I can expose it enough to pull it out.

UPDATE: The nail has to come out so I can put a screw in to hold the bed frame together.

I would use a tapered tool to get under the highest edge and use a plate or something to prevent any more damage to the surface while levering it out.

Done carefully, you can then use screws with some decorative (brass) washers to cover the damage and perhaps leave a better "look".

Something like :

source : https://www.amazon.com/Finishing-Washer-Solid-Countersunk-Washers/dp/B07KY2B7LC

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.

Personally, I would use a nail set tool to drive it in a bit and then use wood filler and then sand after it is dry. It will leave a better aesthetic than digging it out.

If you must remove it, blunt the tip of a nail and use that to drive the short nail all the way through and pull it from the other side once you get enough to clamp onto to pull it out.

You can try what's called a "cats paw nail puller". Used carefully, it can minimize the damage done while pulling out the nail. This is a much smaller pry bar than most other pry bars and is geared towards countersunk nails.

Instead, and if you have access, you can try prying or hammering the pieces of wood apart. Once you have a gap, you can hammer the wood back together, leaving the gap between the wood and the nail head. You might even be able to completely separate the wood pieces and simply drive the nail out at that point.

You can pry and hammer the wood without damaging it if you use a scrap piece between the good wood and your tool, acting as the fulcrum when prying or as the target when hammering.

While not pretty, you can always drill nails out (or at least get the head off). Be sure to use a bit made for metal drilling.

I think I would try a screw extractor first. It's a tool that has a small, reverse-threaded, tapered screw. First you drill a small hole in the head of the offending screw, and then you screw the extractor's screw into it. I found this tool in the automotive tools section at Sears, and it saved me once when I was working on a car.

For a smaller screw, and not having a screw extractor, I'd try hammering the other end of the screw with a nail set to see if I could get the head far enough out to grab with Vise Grips.