  Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph
is a dominating subset $C$ of vertices such that all distinct subsets of
vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood
within $C$. In this paper, we prove that every $k$-iterated line digraph of
minimum in-degree at least 2 and $k\geq2$, or minimum in-degree at least 3 and
$k\geq1$, admits a $(1,\le \ell)$-identifying code with $\ell\leq2$, and in any
case it does not admit a $(1,\le \ell)$-identifying code for $\ell\geq3$.
Moreover, we find that the identifying number of a line digraph is lower
bounded by the size of the original digraph minus its order. Furthermore, this
lower bound is attained for oriented graphs of minimum in-degree at least 2.
