We search for dark matter in the form of axionlike particles (ALPs) in the mass range 5.576741  neV/c2−5.577733  neV/c2 by probing their possible coupling to fermion spins through the ALP field gradient. This is achieved by performing proton nuclear magnetic resonance spectroscopy on a sample of methanol as a technical demonstration of the Cosmic Axion Spin Precession Experiment Gradient (CASPEr-Gradient) Low-Field apparatus. Searching for spin-coupled ALP dark matter in this mass range with associated Compton frequencies in a 240 Hz window centered at 1.348570 MHZ resulted in a sensitivity to the ALP-proton coupling constant of 𝑔ap ≈3 ×10−2  GeV−1. This narrow-bandwidth search serves as a proof-of-principle and a commissioning measurement, validating our methodology and demonstrating the experiment’s capabilities. CASPEr-Gradient Low-Field will probe the mass range from 4.1  peV/c2 to 17  neV/c2 with hyperpolarized samples to boost the sensitivity beyond the astronomical limits.