Ultralight axion-like particles are well-motivated relics that might compose the cosmological dark matter and source anomalous time-dependent magnetic fields. We report on new terrestrial bounds from the Noble And Alkali Spin Detectors for Ultralight Coherent darK matter (NASDUCK) collaboration on the coupling of axion-like particles to neutrons and protons. The detector uses nuclei of noble-gas and alkali-metal atoms and operates in the Spin-Exchange Relaxation-Free (SERF) regime, achieving high sensitivity to axion-like dark matter fields. Conducting a month-long search, we cover the mass range of 1.4×10-12 eV/c2 to 2×10-10 eV/c2 and provide world-leading limits which supersede robust astrophysical bounds, and improve upon previous terrestrial constraints by up to two orders of magnitudes for many masses within this range. These are the first reliable terrestrial bounds reported on the coupling of protons with axion-like dark matter, covering a new and unexplored terrain in its parameter space.