Elastic properties of strongly anisotropic hard cyclic tetramers are studied in the isobaric–isothermal ensemble using Monte Carlo simulations. The tetramers are model molecules consisting of four hard disks of diameter σ with centers forming a square of side d. Recently, it was shown that 2D crystals of hard cyclic tetramers (with anisotropy parameter $\alpha =d/\sigma =1.7$) can form a chiral phase, which is (completely) auxetic, that is, its Poisson's ratio is negative in all directions. Herein, the complete pressure dependence of the Poisson's ratio of the tetramer crystalline phase is presented. These studies show that the auxetic properties occur over a wide pressure range, but the tetramer system is only partially auxetic at low pressures. The Poisson's ratio of the system varies with pressure between $-0.597(3)$ and 0.500(3).