The prediction of new materials with peculiar topological properties is always desirable to achieve new properties and applications. In this work, by means of density functional theory computations, we extend the rule-breaking chemical bonding of planar pentacoordinate silicon (ppSi) into a periodic system: a C2v Ca4Si2 2- molecular building block containing a ppSi center is identified first, followed by the construction of an infinite CaSi monolayer, which is essentially a two-dimensional (2D) network of the Ca4Si2 motif. The moderate cohesive energy, absence of imaginary phonon modes, and good resistance to high temperature indicate that the CaSi monolayer is a thermodynamically and kinetically stable structure. In particular, a global minimum search reveals that the ppSi-containing CaSi monolayer is the lowest-energy structure in 2D space, indicating its great promise for experimental realization. The CaSi monolayer is a natural semiconductor with an indirect band gap of 0.5 eV, and it has rather strong optical absorption in the visible region of the solar spectrum. More interestingly, the unique atomic configuration endows the CaSi monolayer with an unusually negative Poisson's ratio. The rule-breaking geometric structure together with its exceptional properties makes the CaSi monolayer quite a promising candidate for applications in electronics, optoelectronics, and mechanics.