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In this paper, we extend the dynamic spherical cavity expansion model for rate-independent materials developed by Durban and Masri (Int J Solids Struct 41(20):5697-5716, 2004), Masri and Durban (J Appl Mech 72(6):887-898, 2005), and Cohen et al. (J Appl Mech 77(4):041009, 2010) to viscoplastic media. For that purpose, we describe the material behavior with an isotropic Perzyna-type overstress formulation (Perzyna in Q Appl Math 20:321-332, 1963; Adv Appl Mech 9:243-377, 1966) in which the material rate dependence is controlled by the viscosity parameter eta. The theoretical predictions of the cavity expansion model, which assumes that the cavity expands at constant velocity, are compared with finite element simulations performed in ABAQUS/Explicit (Abaqus Explicit v6.13 User's Manual, ABAQUS Inc., Richmond). The agreement between theory and numerical simulations is excellent for the whole range of cavitation velocities investigated, and for different values of the parameter eta. We show that, as opposed to the steady-state self-similar solutions obtained for rate-independent materials (Durban and Masri 2004; Masri and Durban 2005; Cohen et al. 2010), the material viscosity leads to time-dependent cavitation fields and stress relaxation as the cavity enlarges. In addition, we also show that the material viscosity facilitates to model the shock waves that emerge at the highest cavitation velocities investigated, controlling the amplitude and the width of the shock front.