We study the spontaneous symmetry breaking of a superfluid Bose—Fermi mixture in a two-dimensional double-well potential. The mixture is described by a set of coupled Gross—Pitaevskii equations. The symmetry breaking phenomenon is demonstrated in the two-dimensional double-well potential in the mixture. The results are summarized in the phase diagrams of the mixture particle numbers, which are divided into symmetric and asymmetric regions by the asymmetry ratios. The dynamical pictures of the spontaneous symmetry breaking induced by a gradual transformation of the single-well potential into a double-well one are also illustrated. The properties of the quantum degenerate mixture are explored using the realistic parameters for a 40K−87Rb system.