Thermoplastic polyurethanes (TPUs) are candidates for a number of applications where outstanding resilience and ability to dissipate energy under large compressive deformation are needed. TPUs possess complex, heterogeneous structure where chemically distinct segments segregate into hard and soft domains, which pose significant challenges for the molecular level mechanistic understanding of their mechanical properties and associated multiscale modeling and experimentation. In this work, molecular simulations are used to identify the mechanism of mechanical response under large compressive deformation of a common thermoplastic polyurethane comprising 4,4'-diphenylmethane diisocyanate and n-butanediol (hard segment) and poly(tetramethylene oxide) (soft segment), with atomistic resolution. The simulation employs a lamellar stack model constructed by the Interphase Monte Carlo method. This method creates an interfacial zone between hard and soft domains that satisfies both intermolecular packing and intramolecular connectivity constraints. Molecular-level mechanisms responsible for yielding, toughening, and the Mullins effect are reported. We have found several distinct mechanisms for yield and plastic flow in compression, which we categorize as (i) block slip, (ii) fragmentation and restacking, and (iii) block rotation. The activity of these mechanisms depends on the topology of chains in the soft domain and the direction of loading (e.g., parallel or perpendicular to the interface). Further insights regarding the Mullins effect are garnered from cyclic loading at large compressive strains in this family of materials, where mechanisms i and ii were found to be major sources of energy dissipation, while the soft domain was found to be responsible for resilience. Together with our previous study of deformation mechanisms under large tensile strain, the current work expands our understanding of large strain deformation mechanisms for TPUs on the molecular level.