Theories of physics that attempt to reconcile the quantum world with relativity have postulated the existence of additional spatial dimensions: the mathematics of superstring theory gave spacetime a total of ten. However, these theories cause the extra dimensions to wrap themselves up in such a way that they are microscopic - which is not a great help to FTL travel.
Worse: these rather mythical because very-much undetected spatial dimensions do not help to travel FTL: There would just be additional spatial degrees of freedom at each point, but photons would still wander around at c in this more-freedomy micro-space or even, in case those dimensions are unrolled and our 4-D space is a subspace of a larger "bulk" space, in a more freedomy macro-space (though photons do not seem to leave the "brane").
But in any case, my money is on 4D and that's it. I mean, a 4D space with an infinite family of reference frames in each point depending only on relative velocity is already extravagant enough.