High-sensitivity de Haas-van Alphen (DHVA) measurements are reported for pure samples of Ni₃Al in the temperature and magnetic field ranges 0.35<or=T<or=8K and 10<or=H<or=100 kG, respectively. Complex DHVA spectra consisting of up to 20 or more fundamental, harmonic and magnetic breakdown components are interpreted starting with two light and one very heavy sheet of the nonmagnetic Fermi surface. The complex and unusual character of the DHVA spectra associated with the light sheets is shown to arise from the spin-orbit and exchange interactions which are found to yield comparable splittings of the quasidegenerate bands. The ratio of the observed and band-calculated cyclotron masses are close to 2 for the light sheets and as high as 3.4 for the heavy sheet. In addition the ratio of the experimental Curie temperature to that calculated in mean-field band theory in terms of the best available band model is found to be as low as approximately ¹/₁₀. The large mass enhancement and strong suppression of T_c from the mean-field value suggest that enhanced magnetic fluctuations are important in the ground state and thermal properties. A simple description of the effect of both transverse and longitudinal magnetic fluctuations of long wavelength is presented which can account for the observed value of T_c in this material.