We report measurements of the lifetimes of the Λc+, Ξc+ and Ξc0 charm baryons using proton-proton collision data at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3.0  fb−1, collected by the LHCb experiment. The charm baryons are reconstructed through the decays Λc+→pK−π+, Ξc+→pK−π+ and Ξc0→pK−K−π+, and originate from semimuonic decays of beauty baryons. The lifetimes are measured relative to that of the D+ meson, and are determined to be τΛc+=203.5±1.0±1.3±1.4  fs,  τΞc+=456.8±3.5±2.9±3.1  fs,  τΞc0=154.5±1.7±1.6±1.0  fs, where the uncertainties are statistical, systematic, and due to the uncertainty in the D+ lifetime. The measurements are approximately 3–4 times more precise than the current world average values. The Λc+ and Ξc+ lifetimes are in agreement with previous measurements; however, the Ξc0 baryon lifetime is approximately 3.3 standard deviations larger than the world average value.