I was wondering whether a real valued function on the space $C$ of continuous real functions (defined on a closed interval $[a,b]$) i.e. $$ C= \mathcal{C}([a,b];\mathbb{R}^d) $$ which is locally Lipschitz is also Lipschitz in sets as $$ A:=\{ f \in C : || f ||_{\infty} \leq k \} $$ or $$ B:=\{ f \in C : || f -g||_{\infty} \leq k \} $$ for some $k>0$ and some continuous function $g \in C$. As I found out here: Compact sets in uniform norm compactness can't be used.
P.S. With a real valued function on the space $C$ I mean $$ G\colon C \to \mathbb{R}^d $$
Let $V$ be any infinite-dimensional Banach space, $B=B(0,1)\subset V$ the closed unit ball. Take a countably-infinite subset $E\subset B$ such that the distance between distinct points in $E$ is $> 2r>0$. Consider the disjoint closed balls $B(x_i,r)$ centered at the points $x_i\in E$. For each $i$ define a function $f_i: V\to {\mathbb R}$ supported on $B(x_i,r)$ whose Lipshitz constant on this ball equals $i$. Lastly, set $$ f:= \sum_{i=1}^\infty f_i. $$ This function will be locally Lipschitz on $V$ but not Lipschitz on the ball $B(0,1+r)$.