Dijet events in photon-proton collisions in which there is a large pseudorapidity separation, Deltaeta > 2.5 between the two highest E-T jets are studied with the H1 detector at HERA. The inclusive dijet cross sections are measured as functions of the longitudinal momentum fractions of the proton and photon which participate in the production of the jets, x(p)(jets) and x(gamma)(jets) respectively, Deltaeta, the pseudorapidity P separation between the two highest E-T jets, and E-T(gap), the total summed transverse energy between the jets. Rapidity gap events are defined as events in which E-T(gap) is less than E-T(cut), for E-T(cut) varied between jets 0.5 and 2.0 GeV. The fraction of dijet events with a rapidity gap is measured differentially in Deltaeta, x(p)(jets) and x(gamma)(jets). An excess of events with rapidity gaps at low values of E-T(cut) is observed above the expectation from standard photoproduction processes. This excess can be explained by the exchange of a strongly interacting colour singlet object between the jets.