Understanding how cells change shape dynamically under the influence of external and internal forces requires characterisation of the mechanical response of the cytoplasm, the viscous material that fills their interior. The cytoplasm consists of a porous solid phase bathed in a fluid, the cytosol. As the cytoplasm is incompressible, any cellular shape change necessitates redistribution of the cytosol within the cell and its flow rate sets the time-scale for deformation. How the cytoplasmic mechanical response affects cell physiology remains poorly understood. We show that the unique physical properties of the cytoplasm allow cells to sustain cellular-scale pressure gradients over minute time-scales. As a consequence, pressure-driven mechanisms may play a much greater role in cell physiology than currently appreciated.