Yes, no, and yes. First, Paul, in addressing those whom He elsewhere calls “saints,” makes reference to their “carnality.” In I Corinthians 3 Paul bemoans that he must speak to these to whom God’s grace has been given (I Corinthians 1:4) as to the carnal rather than spiritual, describing them as “babes in Christ.” All living Christians deal with carnality, the remaining temptations of the flesh. (Remember that by flesh we do not mean our physical bodies, but rather our sin natures, from the Greek sarx.) We have seen the power of sin broken in our lives. We are new men. But that sin principle is not eradicated. Indeed if we say that it has, if we say that we are without sin, the truth is not in us (I John 1:8). The victory has been won in principle, but there remains a mopping up operation in the great war against our own sin. As such, carnality, sin, sarx, the flesh is still with us until we die.
That said, there are those who would define “carnal Christian” as one who has accepted the saving work of Christ, but whose life is unchanged. Not so. All those who trust in the finished work of Christ are indwelt by the Holy Spirit. Though we all start in different places, and all grow at different rates, we all grow in grace. (Indeed we are changed already when we are given new hearts by the Holy Spirit when He regenerates us.). As John McArthur so ably argued in his book The Gospel According to Jesus, one cannot divide Jesus, accepting Him as Savior while rejecting Him as Lord. That wing of the dispensational church which teaches this doctrine is not just mistaken, but is propagating gross and heinous error. A “Christian” with no change in his life is no Christian at all.
Which brings us to our second “yes.” Here the difficult word isn’t “carnal” but “Christian.” That is, we must be careful not to equivocate on the term. Sometimes when we say “Christian” we mean someone who trusts in the finished work of Christ alone for his salvation, who has peace with the Father and who will spend eternity in paradise. Sometimes, however, what we mean by the term is “Someone who claims to be a Christian.” Because we cannot fully see into anyone’s heart we cannot always distinguish between professors of the Christian faith and possessors of the Christians faith. The word “Christian” can be appropriate in either circumstance, so long as we don’t conflate the differing meanings. There is nothing wrong with using the term Christian for all who profess the faith, nor for using the term for only those who truly possess the faith. There are no carnal Christians in one sense, but there are plenty in the other. Those in this latter category do not need to be assured of their salvation, but need to be called to repentance. A life of unrepentant gross and heinous sin is evidence not of a carnal believer, but a carnal unbeliever.