The quest for planar sp2-hybridized carbon allotropes other than graphene, such as graphenylene and biphenylene sheet, has stimulated substantial research efforts because of the materials' predicted unique mechanical, electronic and transport properties. However, their synthesis remains challenging due to the lack of reliable protocols for generating non-hexagonal rings during the in-plane tiling of carbon atoms. Here, we report the bottom-up growth of ultra-flat biphenylene sheet with periodically arranged four-, six-, and eight-membered rings of sp2 carbon atoms, via an unprecedented on-surface inter-polymer hydrogen fluoride (HF)-zipping reaction. The characterization of this biphenylene sheet by scanning probe methods solves the long-standing controversy over its dielectric or metallic nature. We expect the inter-polymer HF-zipping method to complement the toolbox for the synthesis of other nonbenzenoid carbon allotropes.