The discovery of near-room-temperature superconductivity in the lanthanum hydride LaH₁₀ has revolutionized this research field. However, the need to use diamond anvils for the synthesis of hydride superconductors severely limits the experimental techniques to study these materials. Nuclear magnetic resonance (NMR) is one of the key methods for probing spin systems of superconductors. Here we show how ¹H NMR can be realized in diamond anvil cells to study lanthanum polyhydrides at pressures up to 165 GPa. In the newly discovered superhydride LaH₁₂, we observed a pronounced suppression of the ¹H NMR signal intensity below T_c^onset = 260 K in a magnetic field of 7 T, corresponding to the screening of the radio-frequency pulses. Below the critical temperature, all ¹H NMR characteristics, including the nuclear spin-lattice relaxation rate 1/T₁T, exhibit pronounced features that may be associated with superconductivity. In zero field, the radio-frequency signal transmission shows a pronounced drop below T_c^onset ≈ 267‒279 K, indicating the very beginning of the transition in the most ideal microcrystals. A description of the 1/T₁T data with an exponential form allows the estimation of the superconducting gap Δ(0) = 427‒671 K (corresponding to 36.8‒57.8 meV), and the ratio R_Δ = 2Δ(0)/k_BT_c between 3.76 and 5.16 in the synthesized polyhydride.