A measurement of inclusive differential cross sections for very forward jet production is presented as a function of jet energy in proton-lead collisions. These collisions were recorded with the CASTOR calorimeter of the CMS experiment at the LHC, which measures in the pseudorapidity range $-6.6<\eta<-5.2$. Proton beams with 4 TeV and beams of lead nuclei with an energy of 1.58 TeV per nucleon were used, corresponding to a center-of-mass energy per nucleon pair of $\sqrt{s_{\mathrm{NN}}}=5.02\,$TeV. Cross sections with either the proton (p+Pb) or the ion (Pb+p) travelling towards CASTOR are studied and unfolded to the stable particle level for jets with $p_{\text{T}}\geq3\,$GeV. The ratio p+Pb/Pb+p is also presented. The measurements are compared to various event generators, and discussed in terms of the saturation of gluon densities at low fractional parton momenta. None of the event generators considered in this analysis are able to describe all the spectra in their full energy range, and deviations between data and the models in excess of two orders of magnitude are observed.