Measurements of the centrality and rapidity dependence of inclusive jet production in $\sqrt{s_\mathrm{NN}} = 5.02$ TeV proton--lead ($p$+Pb) collisions and the jet cross-section in $\sqrt{s} = 2.76$ TeV proton--proton collisions are presented. These quantities are measured in datasets corresponding to an integrated luminosity of 27.8 nb$^{-1}$ and 4.0 pb$^{-1}$, respectively, recorded with the ATLAS detector at the Large Hadron Collider in 2013. The $p$+Pb collision centrality was characterised using the total transverse energy measured in the pseudorapidity interval $-4.9 < \eta < -3.2$ in the direction of the lead beam. Results are presented for the double-differential per-collision yields as a function of jet rapidity and transverse momentum ($p_\mathrm{T}$) for minimum-bias and centrality-selected $p$+Pb collisions, and are compared to the jet rate from the geometric expectation. The total jet yield in minimum-bias events is slightly enhanced above the expectation in a $p_\mathrm{T}$-dependent manner but is consistent with the expectation within uncertainties. The ratios of jet spectra from different centrality selections show a strong modification of jet production at all $p_\mathrm{T}$ at forward rapidities and for large $p_\mathrm{T}$ at mid-rapidity, which manifests as a suppression of the jet yield in central events and an enhancement in peripheral events. These effects imply that the factorisation between hard and soft processes is violated at an unexpected level in proton--nucleus collisions. Furthermore, the modifications at forward rapidities are found to be a function of the total jet energy only, implying that the violations may have a simple dependence on the hard parton--parton kinematics.