  Let $G$ be a graph embedded on a surface $S_\varepsilon$ with Euler genus
$\varepsilon > 0$, and let $P\subseteq V(G)$ be a set of vertices mutually at
distance at least 4 apart. Suppose all vertices of $G$ have
$H(\varepsilon)$-lists and the vertices of $P$ are precolored, where
$H(\varepsilon)=\Big\lfloor\frac{7 + \sqrt{24\varepsilon + 1}}{2}\Big\rfloor$
is the Heawood number. We show that the coloring of $P$ extends to a
list-coloring of $G$ and that the distance bound of 4 is best possible. Our
result provides an answer to an analogous question of Albertson about extending
a precoloring of a set of mutually distant vertices in a planar graph to a
5-list-coloring of the graph and generalizes a result of Albertson and
Hutchinson to list-coloring extensions on surfaces.
