Matthew Barrett says yes, but it all depends upon how one defines it. He writes,
Doesn't Calvin's argument imply that man is coerced? Not at all, Calvin replies. Man sins willingly. Yes, it is out of necessity, but not out of compulsion. Such a distinction is one of Calvin's chief points in his treatise against Pighius, who argues that necessitas (necessity) implies coactio (coercion).
However, as Paul Helm explains, for Calvin "it does not follow from the denial of free will that what a person chooses is the result of coercion." Coercion negates responsibility, but necessity is "consistent with being held responsible for the action, and being praised or blamed for it." Therefore, Calvin can affirm that man 'acts wickedly by will, not by compulsion.'"
Does this mean that Calvin does affirm "free will"? If by freedom one means, as Pighius argues, that man's will is in no way determined but that man has the self-power to will good or evil toward God (what is today titled libertarian freedom), so that by his own strength he can equally will either, then free will is rejected by Calvin.
But if by free will one means, as Augustine maintained, that man wills out of voluntary necessity (not coercion) then willful choice can be affirmed. Nevertheless, even if man wills out of necessity it is only a necessity to sin prior to effectual grace. "For we do not say that man is dragged unwillingly into sinning, but that because his will is corrupt he is held captive under the yoke of sin and therefore of necessity wills in an evil way. For where there is bondage, there is necessity." Therefore, the bondage of the will to sin remains and yet such slavery is a voluntary and willful captivity. For example, consider the Devil himself. The Devil can only do evil all of the time and yet he is fully culpable for his actions and commits them voluntarily though out of necessity.
So did Calvin believe in free will? That all depends on the meaning. If by free will one means that the unbeliever is in no way necessitated by sin, but has it in his power to either do good or evil toward God, then the answer is no. But if one means that the unbeliever is in total bondage to sin, sinning willfully yet under necessity (not coercion), making him utterly dependent upon God's irresistible grace to liberate him, then Calvin is your man.
Barrett reminds us that the argument is more nuanced than often presented. Yet, this Wesleyan still can't help but believe that Calvin's notion of free will is ultimately nothing more than "smoke and mirrors."