A search for the rare decay Σ^{+}→pμ^{+}μ^{-} is performed using pp collision data recorded by the LHCb experiment at center-of-mass energies sqrt[s]=7 and 8 TeV, corresponding to an integrated luminosity of 3 fb^{-1}. An excess of events is observed with respect to the background expectation, with a signal significance of 4.1 standard deviations. No significant structure is observed in the dimuon invariant mass distribution, in contrast with a previous result from the HyperCP experiment. The measured Σ^{+}→pμ^{+}μ^{-} branching fraction is (2.2_{-1.3}^{+1.8})×10^{-8}, where statistical and systematic uncertainties are included, which is consistent with the standard model prediction.