Microfluidic sorting of deformable particles finds many applications, for example, medical devices for cells. Deterministic lateral displacement (DLD) is one of them. Particle sorting via DLD relies only on hydrodynamic forces. For rigid spherical particles, this separation is to a great extent understood and can be attributed to size differences: large particles displace in the lateral direction with respect to the flow while small particles travel in the flow direction with negligible lateral displacement. However, the separation of non-spherical deformable particles such as red blood cells (RBCs) is more complicated than that of rigid particles. For example, is it possible to separate deformable particles that have the same size but different mechanical properties? We study deformability-based sorting of same-size RBCs via DLD using an in-house integral equation solver for vesicle flows in two dimensions. Our goal is to quantitatively characterize the physical mechanisms that enable the cell separation. To this end, we systematically investigate the effects of the interior fluid viscosity and membrane elasticity of a cell on its behaviour. In particular, we consider deep devices in which a cell can show rich dynamics such as taking a particular angular orientation depending on its mechanical properties. We have found out that cells moving with a sufficiently high positive inclination angle with respect to the flow direction displace laterally while those with smaller angles travel with the flow streamlines. Thereby, deformability-based cell sorting is possible. The underlying mechanism here is cell migration due to the cell’s positive inclination and the shear gradient. The higher the inclination is, the farther the cell can travel laterally. We also assess the efficiency of the technique for dense suspensions. It turns out that most of the cells in dense suspensions do not displace in the lateral direction no matter what their deformability is. As a result, separating cells using a DLD device becomes harder.