We consider the behaviour of scattering amplitudes and form factors with massive particles in N=4 Maximally Supersymmetric Yang-Mills Theory (SYM) with spontaneously broken gauge symmetry in the planar limit (i.e. on the Coulomb branch of the theory). We demonstrate the non-universality of the infrared asymptotics of such objects. Based on direct three-loop calculations of the Sudakov form factor on the Coulomb branch, a hypothesis is proposed regarding the structure of the infrared asymptotics of scattering amplitudes and form factors with massive particles on external lines at all orders of perturbation theory. Additionally, the structure of the finite parts of scattering amplitudes and form factors with massive particles is discussed using the example of three-point form factors of 1/2-BPS operators in the two-loop approximation.