The proximal point operator for a convex function $f:\mathbb{R}^D\rightarrow \mathbb{R}$ can be defined to be $$ \operatorname{prox}_{\lambda f}(v)=\underset{x}{\operatorname{argmin}}\left(f(x)+\frac{1}{2 \lambda}\|x-v\|_2^2\right). $$ I've read that one can prove $\operatorname{prox}_{\lambda f}$ is contractive for strongly convex $f$ and (firmly) non-expansive in the general convex case with respect to the $\ell_2$ norm.
Could a similar result be shown for an arbitrary $\ell_p$ norm? For instance, could $\operatorname{prox}_{\lambda f}: (\mathbb{R}^D, \|\cdot \|_1) \rightarrow (\mathbb{R}^D, \|\cdot \|_1)$ be shown to be a contraction as well?
For context, I'm referring to this result (Page 131): https://web.stanford.edu/~boyd/papers/pdf/prox_algs.pdf