Local codes specify the minimum height a ceiling must be--this is often 7 feet. If your home's basement ceiling is lower than allowed, you may still be able to convert the basement, but you'll add significantly to the expense, the labor, or both.
It's often cheaper and easier to get the extra space you need by adding on to the house. One option for gaining ceiling height is to have your house lifted off of its foundation so that the foundation walls can be heightened.
You can hire contractors who specialize in lifting and moving houses. This job is not to be taken lightly--no pun intended. When done properly, it;s costly and quite involved.
Another option is to excavate the basement floor. If you need only a few extra inches, this might be a reasonable option. But excavation is a major chore and it is critical that the foundation not be undermined or compromised. You'll definitely need to consult a structural and/or soils engineer.