Solid mechanics problems consisting of materials with different properties are quite common in science and engineering. Particularly, those problems in which one or all the materials are soft and undergo incompressible deformation behaviour are becoming more relevant because of their applications in biomedical engineering [1], soft robotics, and shape morphing [2, 3]. A common characteristic of problems with material interfaces is the discontinuity of normal stresses at the interfaces, which complicates the simulation of solid mechanics problems when one or all of the materials are incompressible. To capture such discontinuities in stresses, the pressure field, which is solved as a Lagrange multiple enforcing the incompressibility constraint [4], must be discontinuous across the elements. This requirement significantly limits the available higher-order finite element spaces that are inf-sup stable and applicable to problems with complex geometries. In this work, we propose a finite element framework based on mixed displacement-pressure formulation [4, 5] that is capable of resolving stress discontinuities accurately. Two inf-sup stable combinations of triangular and tetrahedral elements with element-wise discontinuous pressure are presented along with the Q1/P0 element, and their performance is assessed using several numerical examples and problems of practical interest.