We investigate infinite highly arc transitive digraphs with two additional properties, descendant-homogeneity and Property $Z$. A digraph $D$ is {\itshape {highly arc transitive}} if for each $s \geq 0$ the automorphism group of $D$ is transitive on the set of directed paths of length $s$; and $D$ is {\itshape {descendant-homogeneous}} if any isomorphism between finitely generated subdigraphs of $D$ extends to an automorphism of $D$. A digraph is said to have {\itshape {Property $Z$}} if it has a homomorphism onto a directed line. We show that if $D$ is a highly arc transitive descendant-homogeneous digraph with Property $Z$ and $F$ is the subdigraph spanned by the descendant set of a directed line in $D$, then $F$ is a locally finite 2-ended digraph with equal in- and out-valencies. If, moreover, $D$ has prime out-valency then $F$ is isomorphic to the digraph $\Delta_p$. This knowledge is then used to classify the highly arc transitive descendant-homogeneous digraph of prime out-valency which have Property $Z$.