  We characterize $k-$smoothness of an element on the unit sphere of a
finite-dimensional polyhedral Banach space. Then we study $k-$smoothness of an
operator $T \in \mathbb{L}(\ell_{\infty}^n,\mathbb{Y}),$ where $\mathbb{Y}$ is
a two-dimensional Banach space with the additional condition that $T$ attains
norm at each extreme point of $B_{\ell_{\infty}^{n}}.$ We also characterize
$k-$smoothness of an operator defined between $\ell_{\infty}^3$ and
$\ell_{1}^3.$
