I have a conundrum: If a psychiatric patient who cannot distinguish between dream and reality believes that each night a stranger comes and cuts their arm off, and subsequently asks for their real arm to be cut off in order to prevent the phantasm from doing so, is this an exercise in free will? Or confused with something else?
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.
Your patient can't distinguish between reality and dreams.
In the dream reality his willed and considered decision makes sense; in that there is a rationale behind his decision: he does not want the stranger to cut off his arm every night; perhaps he wants to spite this stranger who is cutting off his arm with such monotonous regularity; in this dream-reality his wilful act appears to be act of will against the phantasm.
And who is to say what happens in his dream world next? For in the morning he may find his arm growing back? Or he might find himself in a forest standing by a Green Knight who cutting off the branch of a tree and placing it on his shoulder turn it into his arm; and then he himself turn into a blue nightingale that the forest into a rose-garden.
In the real reality his decision looks monstrous as he will suffer pain, and torment when and if he returns to real reality.
Dreams have a dream logic that isn't the logic of reality; a proper sense of judgement can judge between realities: the fictional, the dreamlike, the religious and the real.