A search for $C!P$ violation in $Lambda^0_b to p K^-$ and $Lambda^0_b to p pi^-$ decays is presented using a sample of $pp$ collisions collected with the LHCb detector and corresponding to an integrated luminosity of 3.0 fb$^{-1}$. The $C!P$-violating asymmetries are measured to be $A_{mathrm{CP}}^{pK^-} = -0.020 pm 0.013pm 0.019$ and $A_{mathrm{CP}}^{ppi^-} = -0.035 pm 0.017 pm 0.020 $, and their difference $A_{mathrm{CP}}^{pK^-}-A_{mathrm{CP}}^{ppi^-} = 0.014 pm 0.022 pm 0.010$, where the first uncertainties are statistical and the second systematic. These are the most precise measurements of such asymmetries to date.