Surface groups among cubulated hyperbolic and one-relator groups

Let X be a non-positively curved cube complex with hyperbolic fundamental group. We prove that π₁(X) has a non-free subgroup of infinite index unless π₁(X) is either free or a surface group, answering questions of Gromov and Whyte (in a special case) and Wise. A similar result for one-relator groups follows, answering a question posed by several authors. The proof relies on a careful analysis of free and cyclic splittings of cubulated groups.