I have had a few faucets where I had to break the stems to remove the handles, and if the is the solution make sure you have replacement stems before you start. I have also had very few that did not remove with a puller so I would have to see what kind of puller you are using. Try screwing a "hollow head cap screw" into the stem and press the puller's pin against that. It is hardened so it will not bend and will give a better surface to press against. "What then"? You take the faucet apart and repair it.