For me this is a key question that all Christians ought to consider.
The answer, of course, is Yes (obviously): a slave can be Christian...
But we need to consider what this implies with respect to our own understanding of what 'being a Christian' means.
A slave (at least in extreme forms of slavery) can be forced to say or do any kind of evil - on pain of death (or torture). The only means of refusal is to accept death (or torture).
Therefore, a Christian can say or do any kind of evil - up to the point at which death (or torture) would be accepted.
A slave can be prevented from reading scriptures or attending church or receiving sacraments - and yet that slave can be a Christian.
So, given that the slave can be prevented from doing almost anything, and can be compelled to do almost anything; how can a slave be a Christian?
The question forces us to consider Christianity in its essence. By the Fourth Gospel account of being a Christian, we get that the essence is to love Jesus (have faith in him, trust him), hence follow Jesus through death to resurrected life everlasting.
That's it! And that is how and why a slave can be a Christian.
(Note: In practice, a slave can be compelled to do almost any evil, but cannot be prevented from recognising and repenting evil. A slave can be prevented from doing almost anything, but cannot be prevented from praying - nor from knowing and loving Jesus by his own direct personal experience of the Holy Ghost.)
(Further Note: This argument is already (and may become extremely) relevant to Western Christians, more-or-less, sooner-or-later.)