To get an example that is not CAT(0) you just double a finite volume complex hyperbolic manifold with cusps chopped off. The cusp cross section is pi_1-incompressible and its fundamental group is not virtually abelian (it is virtually nilpotent), and hence it cannot be CAT(0).

Actually, as I show in arXiv:math/0509504v3 ANY closed aspherical manifold can be realized as a codimension one incompressible submanifold of a closed aspherical manifold with nonzero simplicial volume. So there are tons of the kind of examples you ask for.

For some reason I cannot add a comment to DC's reply below, so I put it here.
DC, you did not sound sure that you complex hyperbolic example works. :) Incidentally, for such a double there are two ways to show that its simplicial volume is nonzero. One is to use Gromov's result on gluing along amenable subsets (if my memory serves me this was proved in some detail by Kuessner but I could be wrong). What seems to me a better way is to note that the fundamental group of the double is hyperbolic rel cusp cross-section, and that apply recent paper of Mineyev-Yaman that in this case relative hyperbolicity implies nonvanishing of simplicial volume. Neither way is elementary. Examples in my paper mentioned above is more elementary.