Suppose that $X_t$ is an $\mathbb{R}^d$-valued geometric $p$-rough path, satisfying $$ \Vert \mathbf{X}^j_{s,t} \Vert \leq M(t-s)^{\frac{j}{p}}, $$ for some $M>0$ and every $j \in \{1,\dots,p\}$,
Let $Lip(p)$ be the set of Lipschitz functions from $\mathbb{R}^d$ to itself which are $p$-times differentiable and whose $p^{th}$ derivative is Lipschitz. We make this into a complete metric space using the an analogue of the metric found in this paper, that is $$ d(f,g)\triangleq \max\left\{ Lip(f-g),\sup_{x \in \mathbb{R}^d}\|f(x)-g(x)\|_{\mathbb{R}^d} \right\} . $$
Is the function $F:Lip(p)\rightarrow \mathcal{G}^{(p)}(\mathbb{R}^d)$ by $$ f\mapsto \int_0^{\cdot} f(X)dX, $$ continuous? I expect that it is infact Lipschitz but I'm a bit new to the field.
Yes in fact the beauty of rough integrals is continuity in the 1-form $dX$ as well, see theorem 10.41 in "Multidimensional Stochastic Processes as Rough Paths: Theory and Applications"
Here the $V^1,V^2$ are two different choices of $f$.