Two-dimensional conjugated metal–organic frameworks (2D c-MOFs) are emerging as a unique class of electronic materials. However, 2D c-MOFs with band gaps in the Vis-NIR and high charge carrier mobility are rare. Most of the reported conducting 2D c-MOFs are metallic (i.e. gapless), which largely limits their use in logic devices. Herein, we design a phenanthrotriphenylene-based, D2h-symmetric π-extended ligand (OHPTP), and synthesize the first rhombic 2D c-MOF single crystals (Cu2(OHPTP)). The continuous rotation electron diffraction (cRED) analysis unveils the orthorhombic crystal structure at the atomic level with a unique slipped AA stacking. The Cu2(OHPTP) is a p-type semiconductor with an indirect band gap of ≈0.50 eV and exhibits high electrical conductivity of 0.10 S cm−1 and high charge carrier mobility of ≈10.0 cm2 V−1 s−1. Theoretical calculations underline the predominant role of the out-of-plane charge transport in this semiquinone-based 2D c-MOF.