We report the results of a search for axionlike dark matter using nuclear magnetic resonance (NMR) techniques. This search is part of the multifaceted Cosmic Axion Spin Precession Experiment program. In order to distinguish axionlike dark matter from magnetic fields, we employ a comagnetometry scheme measuring ultralow-field NMR signals involving two different nuclei (^{13}C and ^{1}H) in a liquid-state sample of acetonitrile-2-^{13}C (^{13}CH_{3}CN). No axionlike dark matter signal was detected above the background. This result constrains the parameter space describing the coupling of the gradient of the axionlike dark matter field to nucleons to be g_{aNN}<6×10^{-5} GeV^{-1} (95% confidence level) for particle masses ranging from 10^{-22} eV to 1.3×10^{-17} eV, improving over previous laboratory limits for masses below 10^{-21} eV. The result also constrains the coupling of nuclear spins to the gradient of the square of the axionlike dark matter field, improving over astrophysical limits by orders of magnitude over the entire range of particle masses probed.