  Let $X$ be a product of smooth projective curves over a finite unramified
extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has
good reduction and that $X$ has a $k$-rational point. We propose the following
conjecture. The kernel of the Albanese map
$CH_0(X)^0\rightarrow\text{Alb}_X(k)$ is $p$-divisible. When $p$ is an odd
prime, we prove this conjecture for a large family of products of elliptic
curves and certain principal homogeneous spaces of abelian varieties. Using
this, we provide some evidence for a local-to-global conjecture for zero-cycles
of Colliot-Th\'{e}l\`{e}ne and Sansuc (\cite{Colliot-Thelene/Sansuc1981}), and
Kato and Saito (\cite{Kato/Saito1986}).
