Interstitial flow in tumors is a key mechanism leading to cancer metastasis. Tumor growth is accompanied by the development of a leaky vasculature, which increases intratumoral pressure and generates an outward interstitial flow. This flow promotes tumor cell migration away from the tumor. The nature of such interstitial flow depends on the coupling between hydrodynamic conditions and material properties of the tumor, such as porosity and deformability. Here we investigate this coupling by means of a microfluidic model of interstitial flow through a tumor, which is represented by a tumor cell aggregate. For a weak intratumoral pressure, the model tumor behaves as a viscoelastic material of low permeability, which we estimate by means of a newly developed microfluidic device. As intratumoral pressure is raised, the model tumor deforms and its permeability increases. For a high enough pressure, localized intratumoral fracture occurs, which creates preferential flow paths and causes tumor cell detachment. The energy required to fracture depends on the rate of variation of intratumoral pressure, as explained here by a theoretical model originally derived to describe polymer adhesion. Besides the well-established picture of individual tumor cells migrating away under interstitial flow, our findings suggest that intratumoral pressures observed in tumors can suffice to detach tumor fragments, which may thus be an important mechanism to release cancer cells and initiate metastasis.