The Chow group of zero-cycles of an algebraic variety is a mysterious object whose structure depends in a crucial way on the arithmetic properties of the base field. For a certain class of K3 surfaces over a finite extension $k$ of $\mathbb{Q}_p$, we show that if $k/\mathbb{Q}_p$ is unramified then the Chow group of zero-cycles of degree $0$ is a divisible group. On the other hand, we give examples to demonstrate that for ramified extensions $k/\mathbb{Q}_p$, the quotient of the Chow group by its maximal divisible subgroup can be an arbitrarily large finite group.

This is joint work with Evangelia Gazaki.