We report the observation of an anomalous ${}^{27}\mathrm{Al}\ensuremath{-}\mathrm{NMR}$ response of a single grain ${\mathrm{Al}}_{72.4}{\mathrm{Pd}}_{20.5}{\mathrm{Mn}}_{7.1}$ icosahedral quasicrystal at low temperatures. In an external magnetic field of 6 T and upon decreasing temperature, we observe a sharp 100% increase of the resonance linewidth at 2.5 K. No further changes of the linewidth are observed down to 0.05 K. The linewidth enhancement is accompanied by a small but distinct increase of the spin-lattice relaxation rate ${T}_{1}^{\ensuremath{-}1}.$ These anomalies are absent in external fields of 2.5 T and below. Our observations indicate unusual variations in the stability of isolated magnetic moments in a quasiperiodic metallic environment.