Let Gamma be a graph and G be a 2-arc transitive automorphism group of Gamma. For a vertex x is an element of Gamma let G(x)(Gamma(x)) denote the permutation group induced by the stabilizer G(x) of x in G on the set Gamma(x) of vertices adjacent to x in Gamma. Then Gamma is said to be a locally projective graph of type (n, q) if G(x)(Gamma(x)) contains PSLn(q) as a normal subgroup in its natural doubly transitive action. Suppose that Gamma is a locally projective graph of type (n, q), for some n greater than or equal to 3, whose girth (that is, the length of a shortest cycle) is 5 and suppose that G(x) acts faithfully on Gamma(x). (The case of unfaithful action was completely settled earlier.) We show that under these conditions either n = 4, q = 2, Gamma has 506 vertices and G congruent to M-23 or q = 4, PSLn(4) less than or equal to G(x) less than or equal to PGL(n)(4), and Gamma contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2(20) vertices.