  The investigation of s-arc-transitivity of digraphs can be dated back to 1989
when the third author showed that s can be arbitrarily large if the action on
vertices is imprimitive. However, the situation is completely different when
the digraph is vertex-primitive and not a directed cycle. In 2017 the second
author, Li and Xia constructed the first infinite family of G-vertex-primitive
2-arc-transitive examples, and asked if there is an upper bound on s for
G-vertex-primitive s-arc-transitive digraphs w=that are not directed. In 2018
the second author and Xia showed that if there is a largest such value of s
then it will occur when G is almost simple. So far it has been shown that s\leq
2 for almost simple groups whose socle is an alternating group or a projective
special linear group. The contribution of this paper is to prove that s\leq 1
in the case of the Suzuki and the small Ree groups. We give constructions with
s=1 to show that the bound is sharp.
