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This letter presents a cellular solid characterized by extreme tailorability of its mechanical response over a large range of macroscopic strains. The proposed periodic structure is based on a hexagonal chiral lattice topology, modified by introducing transverse curvature in the ligaments, key to the resulting unconventional behavior. The resulting topology allows to exploit different levels of structural hierarchy for tailoring the mechanical behavior of the continuous medium. The capability of the initially curved ligaments to change shape under global deformations is exploited to alter the microstructural motions and - as a result - the macroscopic response. We explore the effect of the additional geometrical design parameter, namely the transverse ligament curvature, on the strain-dependent stiffness of the chiral lattice, evaluating its interplay with the conventional attributes defining the chiral topology and assessing the attainable envelope of mechanical responses. Purposely choosing the parameters permits to tailor the behavior of the chiral lattice, enabling extreme variations in stiffness. The possibility to attain effective negative and zero-stiffness regimes over large compressive strains further exemplifies the potential of this design. These unique characteristics can be used to attain specific wave propagation properties, augment structural damping, control vibrations and noise, and tune the deformation of compliant structures. [ABSTRACT FROM AUTHOR]