Bounded solutions of finite lifetime to differential equations in Banach spaces

Consider a smooth vector field $f\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to {\mathbb{R}}^n$ and a maximal solution $\gamma \text{\hspace{0.17em}⁣}:]a,b[\to {\mathbb{R}}^n$ to the ordinary differential equation $x^{\mathrm{\prime }}=f(x)$. It is a well-known fact that, if $\gamma$ is bounded, then $\gamma$ is a global solution, i.e., $]a,b[=\mathbb{R}$. We show by example that this conclusion becomes invalid if ${\mathbb{R}}^n$ is replaced with an infinite-dimensional Banach space.