  We shall study minimal complex surfaces with $c^2 = 9$ and $\chi=5$ whose
canonical classes are divisible by $3$ in the integral cohomology groups, where
$c_1^2$ and $\chi$ denote the first Chern number of an algebraic surface and
the Euler characteristic of the structure sheaf, respectively. The main results
are a structure theorem for such surfaces, the unirationality of the moduli
space, and a description of the behavior of the canonical map. As a byproduct,
we shall also rule out a certain case mentioned in a paper by
Ciliberto--Francia--Mendes Lopes. Since the irregularity $q$ vanishes for our
surfaces, our surfaces have geometric genus $p_g = 4$.
