We investigate a heavy fermion behaviour and a deformation of an effective potential of the ions due to electron–phonon interaction for the periodic Anderson–Holstein model by using the dynamical mean-field theory combined with the exact diagonalization method. For the strong electron–phonon coupling, the system shows an anomalous heavy fermion behaviour with a large lattice fluctuation and an extreme phonon softening, and then an effective potential for the ions changes from a simple harmonic potential for the non-interacting case to a double-well potential for the strong electron–phonon coupling.