Let $X$ be an infinite-dimensional real Banach space. It is well-known that in this context, the Darboux-integrable maps for a (in general) proper subspace of the Riemann-integrable maps of the form $f:[a,b]\to X$.
Moreover, the Riemann-integrable maps continue to form a subspace of the bounded maps of the form $f:[a,b]\to X$. In a finite-dimensional setting, say $\mathbb{R}$, the Dirichlet Function is an example of a bounded map that fails to be Riemann-integrable.
My question is the following: Does there exist a map of the form $f:[a,b]\to X$ which is bounded but not Riemann-integrable?