The actin cortex is a thin layer of actin filaments and myosin motors beneath the outer membrane of animal cells. It determines the cells' mechanical properties and forms important morphological structures. Physical descriptions of the cortex as a contractile active gel suggest that these structures can result from dynamic instabilities. However, in these analyses the cortex is described as a two-dimensional layer. Here, we show that the dynamics of the cortex is qualitatively different when gel fluxes in the direction perpendicular to the membrane are taken into account. In particular, an isotropic cortex is then stable for arbitrarily large active stresses. If lateral contractility exceeds vertical contractility, the system can either from protrusions with an apparently chaotic dynamics or a periodic static pattern of protrusions.