An experimental study of excitation transfer between certain Zeeman sublevels of the $3^{2}P_{J}$ states in strong magnetic fields is reported for $\mathrm{Na}(3^{2}S_{\frac{1}{2}})\ensuremath{-}{\mathrm{Na}}^{*}(3^{2}P_{J})$ resonant collisions. A major decrease in the $M=\ensuremath{-}\frac{3}{2}\ensuremath{\rightarrow}M=\frac{3}{2}$ transfer rate is observed between 4 and 80 kG. Approximate calculations in the correlation-function framework are performed, and the experimental results and theoretical predictions for the field dependence are in good agreement. Some insights into the mechanisms of the collision process are deduced and the particularly important role of the electronic spin is emphasized. The importance of the symmetry properties of the interaction potential is also directly demonstrated.