Spectra of $K^0_S$ mesons and $\Lambda$ hyperons were measured in p+C interactions at 31 GeV/c with the large acceptance NA61/SHINE spectrometer at the CERN SPS. The data were collected with an isotropic graphite target with a thickness of 4% of a nuclear interaction length. Interaction cross sections, charged pion spectra, and charged kaon spectra were previously measured using the same data set. Results on $K^0_S$ and $\Lambda$ production in p+C interactions serve as reference for the understanding of the enhancement of strangeness production in nucleus-nucleus collisions. Moreover, they provide important input for the improvement of neutrino flux predictions for the T2K long baseline neutrino oscillation experiment in Japan. Inclusive production cross sections for $K^0_S$ and $\Lambda$ are presented as a function of laboratory momentum in intervals of the laboratory polar angle covering the range from 0 up to 240 mrad. The results are compared with predictions of several hadron production models. The $K^0_S$ mean multiplicity in production processes $<n_{K^0_S}>$ and the inclusive cross section for $K^0_S$ production were measured and amount to 0.127 $\pm$ 0.005 (stat) $\pm$ 0.022 (sys) and 29.0 $\pm$ 1.6 (stat) $\pm$ 5.0 (sys) mb, respectively.
$K^0_S$ production cross sections in the [0, 20] mrad polar interval.
$K^0_S$ production cross sections in the [20, 140] mrad polar interval.
$K^0_S$ production cross sections in the [140, 240] mrad polar interval.
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