Does single-layered Aluminium Phosphide exist?
I suspect that it does not.  As far as I know, all trivalent phosphorous is highly pyramidal--or at least not planar. This would rule out any graphene-like structure.

The reasoning that I have seen is based on hybridization. In the second row (third period) the 3s and 3p atomic orbitals are well separated, implying that s and p can not mix and that the outer valence is unhybridized. If bonding is with unhybridized p's, then bonding would occur at right angles.

(Kutzelnigg, W. Angew. Chem., Int. Ed. Engl. 1984, 23, 272.)

(Brown, E. Organometallics 2000, 19, 2208-2214.)