In "Introduction to Smooth Manifolds" (IntSM) by J. M. Lee and "Ordinary Differenial Equations and Dynamical Systems" (ODEDS) by Thomas C. Sideris there is a separate proof for the uniqueness of the ODE solution, whose existence is proved by showing that it is the fixed point of the contraction defined by the Picard iteration.
I don't understand why the uniqueness is not a consequence of the contraction mapping theorem. In an another book, also called, ODEDS by G. Teschl there is no such separate proof for the uniqueness although all these authors mentioned have (essentially) the same proof for the existence.
Any reason why one may doubt about the fixed point being the unique solution of the ODE ? The second author mentioned above writes the following:
The alert reader will have noticed that the solution constructed above is uniquewithin the metric space $X_\delta$, but it is not necessarily unique in $C^0(I_\delta, B_r)$. The next result fills in this gap.
He defines $X_\delta = C^0(I_\delta; B_r)$ as the set of continuous functions from $I_\delta$ to $B_r$. I don't know what he means with $C^0(I_\delta, B_r)$. Notice that the difference is ";" instead of ",".