Based on your successful trial from my speculative comment:
I suspected that if you hit them with a hammer hard enough to get them to move (a bit more in - perhaps 1/8 of an inch) they might then be more willing to pull out (since I expect they are corroded-into-the-pressure-treated-wood for "why they are stuck so hard?")
You say it worked, so ...
I use a nail Jack or puller just as you have.
Most folks that can’t pull the nail are using the puller with the handle down, raise it up bite the nail and pull, this doubles the torque and has never failed me.
You have the shaft of the nail available so the nail Jack is not even needed.
A framing hammer will work also. We don’t use the head but the nail ...