Let $X$ be a compact Polish space and $Y$ be a separable real Banach space. Assume $U \subseteq X \times Y$ is open, bounded in $Y$-norm, and s.t. for any $x \in X$, $\{y \in Y \mid (x,y) \in U\}$ is convex and nonempty. Does it follow that there is a continuous function $f: X \rightarrow Y$ s.t. the graph of $f$ is contained in $U$?

Tagged homotopy-theory since, even though there is no homotopy theory in the formulation of the question, I suspect there might be a homotopy theoretic argument along the lines of "U is a fibration with contractible fibers."