Is the spiritual separation from God for the carnally minded eternal?

Yes. Scripture teaches that to be carnally or fleshly minded is spiritual death, which is separation from God, not annihilation. For those who remain in old Adam, continuously fleshly, that separation is not just for this life but forever. Thus it is eternal.

Les explains from Romans 8:5-6 that there are two types of people: those after the flesh and those after the Spirit. The flesh refers to old Adam, the sin-bent nature controlling a person. To be carnally or fleshly minded is death—spiritual death—which is not annihilation but separation from God. If a person remains continuously carnally minded, still in old Adam, that spiritual separation from God is not just for this life but forever. Les notes that verse 7 begins to describe the carnal mind, and though people may deny it applies to them, it does regardless of how good they consider themselves.

Romans 8:6

For to be carnally minded is death; but to be spiritually minded is life and peace.

What does it mean that to be carnally minded is death?

To be carnally minded means to be fleshly minded, under the control of old Adam, and lacking eternal life. It results in spiritual death, eternal, because the one who does not have the Son does not have life and remains condemned.