In the $k_T$-factorization for exclusive processes, the nontrivial $k_T$-dependence of perturbative coefficients, or hard parts, is obtained by taking off-shell partons. This brings up the question of whether the $k_T$-factorization is gauge invariant. We study the $k_T$-factorization for the case $pi gamma^* to gamma$ at one-loop in a general covariant gauge. Our results show that the hard part contains a light-cone singularity that is absent in the Feynman gauge, which indicates that the $k_T$-factorization is {it not} gauge invariant. These divergent contributions come from the $k_T$-dependent wave function of $pi$ and are not related to a special process. Because of this fact the $k_T$-factorization for any process is not gauge invariant and is violated. Our study also indicates that the $k_T$-factorization used widely for exclusive B-decays is not gauge invariant and is violated.
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