We propose a D-dimensional generalization of 4D biscalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a particular strong-twist limit of γ-deformed N=4 supersymmetric Yang-Mills theory. Similar to the 4D case, the planar correlators of this D-dimensional theory are conformal and dominated by "fishnet" Feynman graphs. The dynamics of these graphs is described by the integrable conformal SO(1,D+1) spin chain. In 2D, it is the analogue of Lipatov's SL(2,C) spin chain for the Regge limit of QCD but with the spins s=1/4 instead of s=0. Generalizing recent 4D results of Grabner, Gromov, Korchemsky, and one of the authors to any D, we compute exactly at any coupling a four-point correlation function dominated by the simplest fishnet graphs of cylindric topology and extract from it exact dimensions of operators with chiral charge 2 and any spin together with some of their operator product expansion structure constants.