I am interested in groups acting properly and cocompactly by isometries on finite products of Gromov-hyperbolic metric spaces. I am mostly interested in the case where the group itself is not virtually a product of Gromov-hyperbolic groups.

Classical examples include irreducible uniform lattices in products or rank $1$ simple Lie groups, and irreducible uniform lattices acting on products of trees. These examples are all $\operatorname{CAT}(0)$.

Question.Are there examples of such groups which are not $\operatorname{CAT}(0)$, or not known to be $\operatorname{CAT}(0)$?

discretegroup (i.e., $\mathrm{Isom}(X)$ has a cocompact lattice, does each $X_i$ admit such an action (i.e., does each $\mathrm{Isom}(X_i)$ admit a cocompact lattice)? $\endgroup$