We search for lepton-number- and baryon-number-violating decays τ-→p¯e+e-, pe-e-, p¯e+μ-, p¯e-μ+, p¯μ+μ-, and pμ-μ- using 921 fb-1 of data, equivalent to (841±12)×106 τ+τ- events, recorded with the Belle detector at the KEKB asymmetric-energy e+e- collider. In the absence of a signal, 90% confidence-level upper limits are set on the branching fractions of these decays in the range (1.8-4.0)×10-8. We set the world's first limits on the first four channels and improve the existing limits by an order of magnitude for the last two channels.