  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.
