Unlike graphene, a hexagonal InP sheet (HInPS) cannot be obtained by mechanical exfoliation from the native bulk InP, which crystallizes in the zinc blende structure under ambient conditions. However, by ab initio density functional theory calculations we found that a slightly buckled HInPS is stable both in pristine form and when doped with Zn atoms; the same occurred for hydrogen-passivated zigzag InP nanoribbons (ZInPNRs), quasi-one-dimensional versions of the quasi-two-dimensional material. We investigated the electronic properties of both nanostructures, in the latter case also in the presence of an external transverse electric field, and the results are compared with those of hypothetical planar HInPS and ZInPNRs. The band gaps of planar ZInPNRs were found to be tunable by the choice of strength of this field, and to show an asymmetric behavior under weak electric fields, by which the gap can either be increased or decreased depending on their direction; however, this effect is absent from slightly buckled ZInPNRs. The binding energies of the acceptor impurity states of Zn-doped HInPS and ZInPNRs were found to be similar and much larger than that of Zn-doped bulk InP. These latter findings show that the reduction of the dimensionality of these materials limits the presence of free carriers.