In this Go position, can black capture the entire length of the white arm?
Yes.  That white chain has only two liberties - A and the space next to A.  If black fills both of them the chain will be captured. None of the surrounding black stones are vulnerable to capture because of their eyes, so white has no defense against this play. 

The space next to A will have no liberties when black plays in it, but that is okay because capturing the white pieces will open up the liberties as black plays their piece. 

White could technically capture the stone at A by playing next to it, but doing so would be pointless because it would use up the other liberty and black would still be able to capture.