In some sub-disciplines of philosophy, a distinction is drawn between free will and autonomy. Free will refers to things a person willfully elects but autonomy refers to things that reflect both rationality and choice. This distinction partially echoes a distinction we find in Aristotle's Nicomachean Ethics about the difference between actions we will to do and actions which follow from choice (which is a more elaborate notion invoking both a wish and rationality and will in the choice).
The example you give is one where this sort of distinction is helpful to make sense of what is going on.
Presumably, the individual in question is choosing (in the simplistic sense) to have their arm cut off. But then as we move into the question of autonomy, we have solid reasons to doubt that this is a choice built on autonomy.
An important missing detail noted by Nick R's comment on your question is that I'm answering this from a certain philosophy of psychology, namely, one where we consider the effects of this phantom experience to be altering the self's rationality but not preventing the self's ability to engage in free actions.
Thus, we both need to answer how we understand psychological problems and the nature of free will with respect to autonomy. There is no single answer to this per se, but what I provide above is what I take to be the most common contemporary theory.