We prove the existence of a global attractor in $\mathcal {S}^{2}_{0}({\Omega })\cap L^{2p-2}({\Omega })$ for a semilinear strongly degenerate parabolic equation in a bounded domain with the homogeneous Dirichlet boundary condition, in which the nonlinearity satisfies a polynomial type condition of arbitrary order and the external force belongs to $L^2(\Omega)$. This global attractor is then shown to have a finite fractal dimension in $L^2(\Omega)$. We also study the existence and exponential stability of the unique stationary solution to the problem.