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 $z$.