Current models for protrusive motility in animal cells focus on cytoskeleton-based mechanisms, where localized protrusion is driven by local regulation of actin biochemistry. In plants and fungi, protrusion is driven primarily by hydrostatic pressure. For hydrostatic pressure to drive localized protrusion in animal cells, it would have to be locally regulated, but current models treating cytoplasm as an incompressible viscoelastic continuum or viscous liquid require that hydrostatic pressure equilibrates essentially instantaneously over the whole cell. Here, we use cell blebs as reporters of local pressure in the cytoplasm. When we locally perfuse blebbing cells with cortex-relaxing drugs to dissipate pressure on one side, blebbing continues on the untreated side, implying non-equilibration of pressure on scales of approximately 10 microm and 10 s. We can account for localization of pressure by considering the cytoplasm as a contractile, elastic network infiltrated by cytosol. Motion of the fluid relative to the network generates spatially heterogeneous transients in the pressure field, and can be described in the framework of poroelasticity.