We have studied proton diffusion and trapping in barium cerate using muon spin relaxation, where positive muons behave as a 'light isotope' of protons. The pure material and samples doped with Sc3+ show trapping at around 200 K near paramagnetic Ce3+ ions. Nd3+ doping (which gives high proton conductivity at high temperatures) shows muon diffusion down to below 100 K. The results obtained also suggest the presence of weak paramagnetism, even in the pure material.