We present an improved determination of the strange quark and antiquark parton distribution functions of the proton by means of a global QCD analysis that takes into account a comprehensive set of strangeness-sensitive measurements: charm-tagged cross sections for fixed-target neutrino–nucleus deep-inelastic scattering, and cross sections for inclusive gauge-boson production and W-boson production in association with light jets or charm quarks at hadron colliders. Our analysis is accurate to next-to-next-to-leading order in perturbative QCD where available, and specifically includes charm-quark mass corrections to neutrino–nucleus structure functions. We find that a good overall description of the input dataset can be achieved and that a strangeness moderately suppressed in comparison to the rest of the light sea quarks is strongly favored by the global analysis.