A　graph　of　order　n　is　p-factor-critical,　where　p　is　an　integer　with　the　same　parity　as　n,　if　the　removal　of　any　set　of　p　vertices　results　in　a　graph　with　a　perfect　matching.　It　is　well　known　that　a　connected　vertex　transitive　graph　is　1-factor-critical　if　it　has　odd　order　and　is　2-factor-critical　or　elementary　bipartite　if　it　has　even　order.　In　this　paper,　we　show　that　a　connected　non　bipartite　vertex-transitive　graph　G　with　degree　k　＞=　6　is　p-factor-critical,　where　p　is　a　positive　integer　less　than　k　with　the　same　parity　as　its　order,　if　its　girth　is　not　less　than　the　bigger　one　between　6　and　k(p-1)+8/2(k-2).