Removing old concrete fence post

Farm jacks are not expensive. I have one, have used it to pull up concrete posts and well established bushes. It's pretty unstoppable, it's capable of lifting a Land Rover so a concreted in fence post is nothing.

You will need something horizontal, e.g. a bolt through a hole in the post. Most have holes.

If you can't get a car Jack and a chain around it then I would dig around it and smash away at with under ground level with a sledge hammer or angle grind it off or even go at it with an SDS drill. It may have metal reinforcing rods, so watch out for these. The only problem with smashing them off is that you may have to change the position of the posts for a new fence. I did once did some out where space was limited and I used a power washer to get around the old concrete. Good luck

If you are replacing the fence you can get 7foot panels if you need to offset the post.

My three-hapenceworth would be leave the post long and rock it about: shove a hose down there to soften the soil.

If you get lucky it will have been put up with postcrete which will probably fall apart.