A search for the rare decay $\Sigma^+ \to p \mu^+ \mu^-$ is performed using $pp$ collision data recorded by the LHCb experiment at centre-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 $\Sigma^+ \to p \mu^+ \mu^-$ branching fraction is $(2.2\,^{+\,1.8}_{-\,1.3})\times 10^{-8}$, where statistical and systematic uncertainties are included, which is consistent with the Standard Model prediction.