Transverse momentum spectra of charged pions, kaons, and protons are measured in proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$ with the CMS detector at the LHC. The particles, identified via their energy loss in the silicon tracker, are measured in the transverse momentum range of ${p}_{\mathrm{T}}\ensuremath{\approx}0.1--1.7\text{ }\text{ }\mathrm{GeV}/c$ and rapidities $|y|<1$. The ${p}_{\mathrm{T}}$ spectra and integrated yields are compared to previous results at smaller $\sqrt{s}$ and to predictions of Monte Carlo event generators. The average ${p}_{\mathrm{T}}$ increases with particle mass and charged particle multiplicity of the event. Comparisons with previous CMS results at $\sqrt{s}=0.9$, 2.76, and 7 TeV show that the average ${p}_{\mathrm{T}}$ and the ratios of hadron yields feature very similar dependences on the particle multiplicity in the event, independently of the center-of-mass energy of the pp collision.