TL/DR: The theology concerning the existence of a “carnal Christian” holds that a person can realistically come to true saving faith in Christ and continue to live free of repentance and sanctification for the full remainder of his or her life. This paper examines Romans 7:14 and 8:1–14 to demonstrate that Romans knows no such concept. In Romans 7:14, Paul identifies even himself as one who struggles with the flesh. In Romans 8:1–14, there are only two kinds of people in view: the natural and the spiritual. It makes no mention of a middle category.
For a more broad-based overview of “carnal Christian” theology, see GotQuestions.org, which explains that “The key thing to understand is that while a Christian can be, for a time, carnal, a true Christian will not remain carnal for a lifetime.”
This is adapted from a paper I submitted in a Romans exegesis class. I have gone through here to make sure anything appearing in Greek script gets transliterated or translated appropriately, but it is still very technical. Feel free to leave a comment!