Novel three-dimensional (3-D) structures based on sp2 boron nitride schwarzite (BN-schwarzite) with octagon rings are proposed for the first time by first-principles study. The structural stability, and mechanical and electronic properties of three typical BN-schwarzites, that is BN-192P, BN-6.82P and BN-6.82D, are investigated. It is found that the unique cavity structure of BN-schwarzite gives rise to a specific surface area (SSA) as high as 2800 m2 g-1 and intrinsic large pores of 5-13 Å. All BN-schwarzites studied are good ductile materials with Poisson's ratio as high as 0.35, and dynamically stable. An unusually wide band gap higher than 4 eV is observed which cannot be achieved by the existing carbon schwarzite (C-schwarzite). Furthermore, a high lithium diffusion coefficient is obtained in BN-schwarzite, which is estimated to be 30 times more than that of graphene at room temperature. Our theoretical results would motivate further experimental work related to the synthesis and characterization of BN-schwarzite. Such boron nitride structures with porosity and large surface areas may hold great promise in the fields of lithium ion batteries, molecular sieving, and environmental remediation.