  We characterize $k-$smoothness of bounded linear operators defined between
infinite-dimensional Hilbert spaces. We study the problem in the setting of
both finite and infinite-dimensional Banach spaces. We also characterize
$k-$smoothness of operators on some particular spaces, namely
$\mathbb{L}(\mathbb{X},\ell_{\infty}^n),~\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$
where $\mathbb{X}$ is a finite-dimensional Banach space and $\mathbb{Y}$ is a
two-dimensional Banach space. As an application, we characterize extreme
contractions on $\mathbb{L}(\ell_{\infty}^3,\mathbb{Y}),$ where $\mathbb{Y}$ is
a two-dimensional polygonal Banach space.
