Does the Peano existence theorem apply to continuous vector fields on smooth manifolds?
For locally Lipschitz vector fields on smooth manifolds, the Picard-Lindelöf theorem can be applied in local charts to obtain local solutions, and then the uniqueness portion of this theorem guarantees that the local solutions agree where they overlap and thus can patch together to form global solutions.
The Peano theorem guarantees only existence and not uniqueness. Thus I think there is no guarantee that local solutions agree where they overlap, and thus no guarantee that they can patch together to form global solutions.
Is it possible to "choose" particular local solutions in charts so that they do agree where they overlap, and thus patch together to form global solutions?
Edit: As John B pointed out with his answer, my original question wasn't worded very well since the Peano theorem is a local statement. Here is a modified question which captures the essence of what I want to know.
Suppose we have a continuous vector field on a compact manifold. Is this vector field complete, in the sense that any integral curve has maximal interval of existence equal to all of $\mathbb{R}$?
The reason is simple: the theorem is of local nature since it only says that a solution exists in some open neighborhood of the initial time: because of this a single chart is sufficient.
It is not really completely as you describe what happens in the case of the Picard-Lindelöf theorem because the charts could get smaller and smaller where the solution is moving to. The correct argument needs to include the usual final step when you show that a solution can be extended unless it reached the boundary.
On your last question. The word "global" perhaps is used in some other sense than the usual: that the solutions are defined for all time in the real line. Note that in $\mathbb R^n$ all solutions for a given initial condition have the same maximal interval, no matter if the solution is unique or not. Perhaps this is what you are referring to. I don't know whether this is also true on a smooth manifold, but I would conjecture that there are counterexamples.