Let $B$ be a Banach space and $f : [0,+\infty)\times B \to B$ be a continuous function which is Lipschitz continuous in the second argument with Lipschitz constant $L$ (which does not depend on the first argument). By Picard-Lindelof, there is a unique function $x : [0,+\infty) \to B$ such that $x(0) = \mathbf{0}$ and for all $t\in [0,+\infty)$, $x'(t) = f(t,x(t))$. Define the family of functions $Euler_h : [0,+\infty) \to B$ to be the result of linear interpolation between the points obtained by the Euler method on $f$ with initial value $\mathbf{0}$ and step size $h$. Does it follow that $Euler_h$ converges compactly to the unique solution $x$ as $h$ goes to 0 from the right? If yes, is an explicit rate known?

The results I have found are only for the points provided by the Euler method, and not for the approximating functions obtained from them.