  For a quasi-compact K\"ahler manifold $U$ endowed with a nilpotent harmonic
bundle whose Higgs field is injective at one point, we prove that $U$ is
pseudo-algebraically hyperbolic, pseudo-Picard hyperbolic, and is of log
general type. Moreover, we prove that there is a finite unramified cover
$\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ so that any
projective compactification of $\tilde{U}$ is pseudo-algebraically hyperbolic,
pseudo-Picard hyperbolic and is of general type. As a byproduct, we establish
some criterion of pseudo-Picard hyperbolicity and pseudo-algebraic
hyperbolicity for quasi-compact K\"ahler manifolds.
