  Let $D$ be a strongly connected digraph.
  An arc set $S$ of $D$ is a \emph{restricted arc-cut} of $D$ if $D-S$ has a
non-trivial strong component $D_{1}$ such that $D-V(D_{1})$ contains an arc.
The \emph{restricted arc-connectivity} $\lambda'(D)$ of a digraph $D$ is the
minimum cardinality over all restricted arc-cuts of $D$. A strongly connected
digraph $D$ is \emph{$\lambda'$-connected} when $\lambda'(D)$ exists. This
paper presents a family $\cal{F}$ of strong digraphs of girth four that are not
$\lambda'$-connected and for every strong digraph $D\notin \cal{F}$ with girth
four it follows that it is $\lambda'$-connected. Also, an upper and lower bound
for $\lambda'(D)$ are given.
