Let $E$ be a $\mathbb R$-Banach space, $$d(x,y):=\min(1,\left\|x-y\right\|_E)\;\;\;\text{for }x,y\in E,$$ $\Omega\subseteq E$ be open, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:\Omega\\x\:\ne\:y}}\frac{|f(x)-f(y)|}{d(x,y)}\;\;\;\text{for }f:\Omega\to\mathbb R$$ and$^1$ $$\left\|f\right\|_{C^1(\Omega)}:=\left\|f\right\|_\infty+\left\|{\rm D}f\right\|_\infty\;\;\;\text{for Fréchet differentiable }f:E\to\mathbb R.$$
If $f:E\to\mathbb R$ is Fréchet differentiable, are we able to show that $|f|_{\operatorname{Lip}(d)}\le2\left\|f\right\|_{C^1(\Omega)}$?
I'm aware of several related results. For example, if $f:E\to\mathbb R$ is Fréchet differentiable, then it is locally Lipschitz continuous at $x\in\Omega$ with Lipschitz constant $\left\|{\rm D}f(x)\right\|_{E'}+\varepsilon$ for all $\varepsilon>0$ for all $x\in\Omega$. On the contrary, if $f$ if Fréchet differentiable at $x\in\Omega$ and Lischitz continuous at $x$ with constant $c$, then $\left\|{\rm D}f(x)\right\|_{E'}\le c$.
$^1$ As usual, if $E$ is a set and $F$ is a normed $\mathbb R$-vector space, then $\left\|f\right\|_\infty:=\sup_{x\in E}\left\|f(x)\right\|_F$ for $f:E\to F$.