Can an unmarried mother move out of Nevada with her child against the will of the father

What is the status of the court order regarding custody, if any?
Has paternity been established with a DNA test?

How old is the child ?

If there is a court order giving the father partial custody and visitation then you cannot move the child if it will not allow the father to exercise his court ordered visitation. If you do, the father can have the mother cited for contempt of court and have the child brought back to NV and the mother might even lose custody.

If there are no court orders then the mother can move, but the father can go to court in NV to get custody before the child can establish residency in another state. If the father does this he can be awarded custody and bring the child back to NV.

Whether the mother and father are married has little bearing on this. The important thing is that a father has legal rights as a parent.

Bottom line is, if you leave the state against the father's wishes without court approval, then the father can get a court order to bring the child back and you could lose custody

... but the father can go to court in NV to get custody before the child can establish residency in another state. ...

Technically, it's the child's "home state" under UCCJEA, not child's "residency". It takes physical presence for six months.