  We introduce the notion of weakly K\"ahler hyperbolic manifold which
generalizes that of K\"ahler hyperbolic manifold given in the early '90s by M.
Gromov, and establish its basic features. We then investigate its spectral
properties and show a spectral gap result (on a suitable modification). As
applications, we prove that weakly K\"ahler hyperbolic manifolds are of general
type and we study the geometry of their subvarieties and entire curves,
verifying -- among other things -- various aspects of the Lang and the
Green--Griffiths conjectures for this class of manifolds.
