  It is shown that the maximal size of a $2$-arc in the projective Hjelmslev
plane over $\mathbb{Z}_{25}$ is $21$, and the $(21,2)$-arc is unique up to
isomorphism. Furthermore, all maximal $(20,2)$-arcs in the affine Hjelmslev
plane over $\mathbb{Z}_{25}$ are classified up to isomorphism.
