By dimensional reduction of the 3D motif of Cs₂AgBiBr₆, a lead-free 2D hybrid double perovskite, (i-PA)(2)CsAgBiBr₇ (1, i-PA=isopentylammonium), was successfully designed. It adopts a quantum-confined bilayered structure with alternating organic and inorganic sheets. Strikingly, the unique 2D architecture endows it highly anisotropic nature of physical properties, including electric conductivity and optical absorption (the ratio alpha(b)/alpha(c)=1.9 at 405 nm). Such anisotropy attributes result in the strong polarization-sensitive responses with large dichroic ratios up to 1.35, being comparable to some 2D inorganic materials. This is the first study on the hybrid double perovskites with strong polarization sensitivity. A crystal device of 1 also exhibits rapid response speed (ca. 200 mu s) and excellent stabilities. The family of 2D hybrid double perovskites are promising optoelectronic candidates, and this work paves a new pathway for exploring new green polarization-sensitive materials.