Answer
North pole and many points on circles around the south pole, see explanations.
Work Step by Step
We assume the directions North and South originated from single points in the north and south poles.
If she uses a compass and starts from the north pole, going south would be along any direction,
walk 1 mile, turn east and walk (this would be walking in a circle of 1 mile radius around the north pole,
finally when she walks 1 mile north, she will return to the north pole, the starting point.
Now think about the south pole, if we draw of circle around the south pole and let the circumference be 1 mile,
we can start anywhere north of that circle 1 mile away so that walk 1 mile south (to the circle), 1 mile east
(around the circle 1 full turn), and 1 mile north will return back to the starting point.
We can also draw smaller circles such that 1 mile will complete multiple full turns and the result will be the same.
