Does every non-compact manifold admit an incomplete vector field?

Question

I know that every vector field on a compact manifold is complete. The question of whether every non-compact manifold admits an incomplete vector field seems to follow naturally. I'd hazard a guess that the answer is no, but I can't think of an appropriate non-compact space to use as an example. Could someone give me a hint as to how to think of the right space, or inform me that my intuition is wrong? Many thanks in advance!