A complex projective variety is Brody hyperbolic if there are no non-constant holomorphic maps from the complex plane to the variety. We prove that a cyclic cover of a smooth complex projective variety branched along a generic small deformation of a sufficiently large multiple of a Brody hyperbolic ample divisor is Brody hyperbolic. We also show the hyperbolicity of the complements of those branch divisors. These results produce lots of examples of hyperbolic
cyclic covers and complements.

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