We report β-NMR investigations of polarized ⁸Li implanted in thin Pb and Ag/Nb films. At the critical superconducting temperature, we observe a singular peak in the spin relaxation rate in small longitudinal magnetic fields, which is attributed to unexpected slow fluctuations in the superconducting order parameter. The peak is several orders of magnitude larger than the prediction based on the enhancement of the dynamic electron spin susceptibility by superconducting fluctuations. The observed peak in 1 / T₁ is rapidly suppressed in a small magnetic field, indicating that it is due to remarkably slow diamagnetic fluctuations which are undetectable with conventional NMR.