The weakly nonlinear evolution of linearly stable perturbations in Hall magnetohydrodynamic plasmas is investigated by the method of strained coordinates. Unperturbed self-similar equilibrium solutions for Z-pinches with time- dependent total current $I\,{\sim}\, t^S$, are subjected to sausage perturbations in the lossless short scale approximation in which the temporal and spatial variation of basic state is treated as slow. The experimentally observed enhanced stability of the Z-pinches previously explained by the linear stability of the core in the interior of the unperturbed pinch is supported by the present nonlinear analysis. The nonlinear evolution results in an axially periodic series of wave breakings, which are confined to a thin shell around the stable pinch core. The wave breaking time may reach the levels of the pinch lifetime in the vicinity of the near-axis branch of the boundary neutral curve, where the inverse Hall parameter is less than unity and the Hall effects are most significant.