Following Deane Yang, the answer is a definite yes: the map in question is a global diffeo, provided that (a) it is `locally invertible': i.e. its derivative is everywhere invertible, and (b) the domain and range are compact, simply connected, without boundary.
Proof: the map must be a covering map (``stack of records theorem'' -- see for example Guillemin and Pollack). But a covering map between simply connected spaces is an isomorphism -- here a diffeo.
To make this `non-contrived' take domain and range to be the two-sphere.
