For the first time at LHC energies, the forward rapidity gap spectra from proton-lead collisions for both proton and lead dissociation processes are presented. The analysis is performed over 10.4 units of pseudorapidity at a center-of-mass energy per nucleon pair of Formula Presented, almost 300 times higher than in previous measurements of diffractive production in proton-nucleus collisions. For lead dissociation processes, which correspond to the pomeron-lead event topology, the epos-lhc generator predictions are a factor of 2 below the data, but the model gives a reasonable description of the rapidity gap spectrum shape. For the pomeron-proton topology, the epos-lhc, qgsjet ii, and hijing predictions are all at least a factor of 5 lower than the data. The latter effect might be explained by a significant contribution of ultraperipheral photoproduction events mimicking the signature of diffractive processes. These data may be of significant help in understanding the high energy limit of quantum chromodynamics and for modeling cosmic ray air showers.