The inclusive production of $\Lambda$ hyperons in proton-proton collisions at $\sqrt{s}=3.18\mathrm{GeV}$ was measured with HADES at the GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt. The experimental data are compared to a data-based model for individual exclusive $\Lambda$ production channels in the same reaction. The contributions of intermediate resonances such as $\Sigma \left(1385\right)$, ${\Delta }^{++}$, or ${\text{N∗}}^{}$ are considered in detail. In particular, the result of a partial-wave analysis accounts for the abundant $\mathrm{pK}{}^{}\Lambda$ final state. Model and data show a reasonable agreement at midrapidities, while a difference is found for larger rapidities. A total $\Lambda$ production cross section in $p+p$ collisions at $\sqrt{s}=3.18\mathrm{GeV}$ of $\sigma (p+\mathrm{p\to \Lambda }+X)=207.3\pm 1.3{}_{-7.3}^{+6.0}(\mathrm{stat}.)\pm 8.4(\mathrm{syst}.){}_{-0.5}^{+0.4}(\mathrm{model}.)\mathrm{\mu b}$ is found.