If you stay in business long enough, you will come across a client that is hard to please. No matter what you try, this person will not be happy. This person will try to get you to do work for free or at an unreasonable price. Don’t ever back down on your price. There will be other customers that will be happy to work with you at the rate you have set. Use this as a learning experience and keep marketing for new clients.
Here’s a video with some tips about dealing with difficult clients: