Are there any examples in the Bible of sinners being forgiven without evidence of repentance?
No, there is no evidence of such...there can't be, for such is a contradiction against God's plan for salvation.
I think that it needs to really be understood that being of God requires repentance, denial of self, and total submission (or surrender) to the Lord. Unfortunately, many seem to enjoy the idea of being saved without their part of the deal.
Repentance is needed before we can turn, for we need to realize that we are sinners. So we need to repent of our sins. But after the repentance, we need to also stop committing these sins. Granted, we do tend to slip on occasion, but that's where the Lord's grace steps in, for when we get hit with the guilt, realize our sin, and repent again, He again forgives us for it and helps us to become strong against it.
The problem with not repenting is that to do so is to say that we don't have sin, but just want to go to Heaven. That doesn't work. Those who think this also often figure that when they accept Christ, not only are they forgiven, but that their sin is then justified. But God doesn't justify sin, He justifies the sinner. Sin is poison. By justifying sin is to say that it's alright to continue to do it...but it's not! And one of the great many problems among many who claim the name of Christ today is unrepentant sin.
Keep in mind that Jesus' ministry was "Repent, for the Kingdom of God is at hand!" If one is to even SEE the Kingdom of God, they must first repent and be born from above, which comes with baptism.
But to answer your question, if one recognizes the need for forgiveness, the hopes is that they'll repent of them. If they don't, then, according to James, they are saying that they have no sin, and thus calling God a liar.