$k$-smoothness on polyhedral Banach spaces

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 its norm at each extreme point of $B_{\ell _{\infty }^{n}}.$ We also characterize $k$-smoothness of an operator from $\ell _{\infty }^3$ to $\ell _{1}^3.$