  In this work we consider arc criticality in colourings of oriented graphs. We
study deeply critical oriented graphs, those graphs for which the removal of
any arc results in a decrease of the oriented chromatic number by $2$. We prove
the existence of deeply critical oriented cliques of every odd order $n\geq 9$,
closing an open question posed by Borodin et al. (Journal of Combinatorial
Theory, Series B, 81(1):150-155, 2001). Additionally, we prove the
non-existence of deeply critical oriented cliques among the family of circulant
oriented cliques of even order.
