  Let $X$ be a $K3$ surface over a $p$-adic field $k$ such that for some
abelian surface $A$ isogenous to a product of two elliptic curves, there is an
isomorphism over the algebraic closure of $k$ between $X$ and the Kummer
surface associated to $A$. Under some assumptions on the reduction types of the
elliptic curve factors of $A$, we prove that the Chow group $A_0(X)$ of
zero-cycles of degree $0$ on $X$ is the direct sum of a divisible group and a
finite group. This proves a conjecture of Raskind and Spiess and of
Colliot-Th\'{e}l\`{e}ne and it is the first instance for $K3$ surfaces when
this conjecture is proved in full. This class of $K3$'s includes, among others,
the diagonal quartic surfaces. In the case of good ordinary reduction we
describe many cases when the finite summand of $A_0(X)$ can be completely
determined.
  Using these results, we explore a local-to-global conjecture of
Colliot-Th\'{e}lene, Sansuc, Kato and Saito which, roughly speaking, predicts
that the Brauer-Manin obstruction is the only obstruction to Weak Approximation
for zero-cycles. We give examples of Kummer surfaces over a number field $F$
where the ramified places of good ordinary reduction contribute nontrivially to
the Brauer set for zero-cycles of degree $0$ and we describe cases when an
unconditional local-to-global principle can be proved, giving the first
unconditional evidence for this conjecture in the case of $K3$ surfaces.
