It is a well known theorem of Anderson that any vector bundle over a negative sectional curvature Riemannian manifold admits a metric of negative sectional curvature.

Q: Are there examples of complete negative sectional curvature manifolds of the form $X \times Y$, for $Y$ closed and $X$ non-compact and not simply connected?

It should also be noted that X being non-compact is essential, as a theorem of Preissmann precludes negative sectional curvature metrics on compact products.