Preservation of the index of p-adic linear operators under compact perturbations

In this paper it is proved that the index of a Fredholm operator between p- adic Banach spaces is preserved under compact perturbations. A case of spec ial interest is provided when the ground field is nonspherically complete. In this case the classical techniques are no longer valid and the relation between the kernels of a Fredholm operator and that of a small compact pert urbation turn out to be in general much richer than in the complex context.