  We give an explicit lower bound, in terms of the distance from the boundary,
for the Kobayashi metric of a certain class of bounded pseudoconvex domains in
$\mathbb{C}^n$ with $\mathcal{C}^2$-smooth boundary using the regularity theory
for the complex Monge--Ampere equation. Using such an estimate, we construct a
family of unbounded Kobayashi hyperbolic domains in $\mathbb{C}^n$ having a
certain negative-curvature-type property with respect to the Kobayashi
distance. As an application, we prove a Picard-type extension theorem for the
latter domains.
