Suppose $X$ and $Y$ are Banach spaces and $U\subseteq X$, $V\subseteq Y$ are open subsets. Let $f:U\to V$ be bijective and continuously Fréchet differentiable with derivative $Df:U\to\mathcal{L}(X;Y)$. Assume furthermore that the inverse function $f^{-1}:V\to U$ is Fréchet differentiable with derivative $Df^{-1}:V\to\mathcal{L}(Y;X)$.
Under these assumptions, is it true that $Df^{-1}$ is automatically continuous in $V$?
I've tried to write $$Df^{-1}(y)-Df^{-1}(y_0)=Df\left(f^{-1}(y)\right)^{-1}-Df\left(f^{-1}(y_0)\right)^{-1}=Df\left(f^{-1}(y)\right)^{-1}\left(Df\left(f^{-1}(y_0)\right)-Df\left(f^{-1}(y)\right)\right)Df\left(f^{-1}(y_0)\right)^{-1}.$$ The question hence seems to boil down to the question whether or not $$\left\Vert Df(x)^{-1}\right\Vert_{\mathcal{L}(Y;X)}$$ remains bounded as $x\to x_0\in U$. I suspect that some kind of Neumann series argument is required but I couldn't come up with something so far.