Not necessarily... In the example you give - a ball in three dimensions with Euclidean metric - there are continuous families of periodic orbits of all numbers of reflections $n\geq 2$ (polygons and stars on planes passing through the origin).  But a typical perturbation of the metric would break the symmetry and leave only a finite number of periodic orbits for each $n$.  So most of the original periodic orbits are unstable in this sense.