Measurements are reported of the temperature dependence of EPR lifetime broadening in the spectrum of Gd³⁺ ions in tetragonal sites in CaF₂ and of dielectric and anelastic relaxation in similar specimens. Concentrations ranged from 0.01 to 0.27 mol per cent GdF₃, and the crystals were annealed in flowing He plus HF at various temperatures. The strongest relaxation modes (R_I) arising from pairs formed from Gd³⁺ ions plus trapped F^- interstitials could be identified in both the dielectric (T_1u) and the anelastic (E_g) spectra. The rate constants are ln tau ₀=-32.83+or-0.24 ( tau ₀ measured in seconds) and Q=0.44+or-0.005 eV, and are to be associated with jumps of the F^- interstitials originating at the nearest-neighbour sites to the Gd³⁺ ions. A low-temperature relaxation (Q approximately 0.2 eV) was observed in the anelastic spectrum with trigonal symmetry and also in the dielectric spectrum. It was found to consist of several components. It is argued contrary to earlier conclusions, that this relaxation is not an additional mode of the same centre giving rise to R_I. A higher temperature relaxation (Q approximately 1.1 eV) was seen only in the anelastic spectrum, and was approximately isotropic. It too cannot be identified on the basis of present information.