  We prove that a cyclic cover of a smooth complex projective variety is Brody
hyperbolic if its branch divisor is a generic small deformation of a large
enough multiple of a Brody hyperbolic base-point-free ample divisor. We also
show the hyperbolicity of complements of those branch divisors. As an
application, we find new examples of Brody hyperbolic hypersurfaces in
$\mathbb{P}^{n+1}$ that are cyclic covers of $\mathbb{P}^n$.
