Take a very general hypersurface $X$ of sufficiently high degree in $\mathbb{P}^n$. Then it is well known that $X$ is Kobayashi hyperbolic (this was proven by Siu, if I remember correctly).
On the other hand, Lefschetz hyperplane theorem says that if $n \geq 4$ the natural map
$\pi_1(X) \to \pi_1(\mathbb{P}^n)$
is an isomorphism. Then $X$ is simply connected, in particular its fundamental group is amenable and so $X$ cannot be Kahler hyperbolic.