On a practical day to day level, of course we do in the sense that we have personal volition that we can be held accountable for and are in control of.
And in my experience this is where the problem comes up with the way religious people want to discuss it. If there's an all-knowing all-powerful god that created the universe in the way most Christians (and a lot of other religions) believe, then ultimate responsibility rests at His feet. So they need another layer of free will, something that lets them say "Yes, god knew exactly what would happen from the moment He created the universe and he could have created it so that things happened differently BUT this is still your fault and so it's fair that you're going to burn in hell".
Obviously this is a tough thing for them to define, and the most I've ever seen them do is essentially "we have free will because god gave us free will" without any attempt to show that there's a mechanism for it or anything.
Well I think the process can have some quantum noise in it .. nothing in this universe is strictly deterministic .. and brain seems to contain actions small enough for it to have some effect. And even if brain contained strong anti-noise measures, the way our digital computers do, you still have noise at the inputs ..
Anyway .. people can behave quite logically, if they want to, so I guess there is not that much noise .. and obviously, noise isn't free will.
Yeah, the "is the universe deterministic" thing is a red herring. Throwing some randomness into the mix does not generate the kind of free will that people mean when they have this discussion. "I did this because it was a chain of unbroken causality from the beginning of everything" and "I did this partly because of causality and also there were some random variables" aren't different in any significant way because neither involves a choice.
The absolute definition of free will belongs to our Creator alone, but through Christ Jesus we are blessed with our Creator's longing to share that absolute with us.
See, this is a good example of a religious person making it clear they're not going to properly define terms. It's all going to be "we have free will because god".
The best way to conceptualize human free will is to look at some one who is alive and some one who is dead.
That doesn't seem like a useful way to answer this question.
Anyway, there are two types of free will. The everyday one that boils down to "are you responsible for what you just did", which obviously exists in the only way that could possibly matter to us in the real world... and then the free will that religion needs to exist in order to justify hell, which cannot be defined in a way that makes sense and doesn't seem to have any logical space to exist in.
So to answer the question in the thread title... no. Not in the way people who want to debate free will would need.