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The investigation of s-arc-transitivity of digraphs can be dated back to 1989when the third author showed that s can be arbitrarily large if the action onvertices is imprimitive. However, the situation is completely different whenthe digraph is vertex-primitive and not a directed cycle. In 2017 the secondauthor, Li and Xia constructed the first infinite family of G-vertex-primitive2-arc-transitive examples, and asked if there is an upper bound on s forG-vertex-primitive s-arc-transitive digraphs w=that are not directed. In 2018the second author and Xia showed that if there is a largest such value of sthen it will occur when G is almost simple. So far it has been shown that s\leq2 for almost simple groups whose socle is an alternating group or a projectivespecial linear group. The contribution of this paper is to prove that s\leq 1in the case of the Suzuki and the small Ree groups. We give constructions withs=1 to show that the bound is sharp.

Vertex-primitive s-arc-transitive digraphs admitting a Suzuki or Ree group