The flexible photodetector is viewed as a research hotspot for numerous advanced optoelectronic applications. Recent progress has manifested that lead-free layered organic-inorganic hybrid perovskites (OIHPs) are highly attractive to engineering flexible photodetectors due to the effective overlapping of several unique properties, including efficient optoelectronic characteristics, exceptional structural flexibility, and the absence of Pb toxicity to humans and the environment. The narrow spectral response of most flexible photodetectors with lead-free perovskites is still a big challenge to practical applications. In this work, we demonstrate the flexible photodetector based on a novel (to our knowledge) narrow-bandgap OIHP of (BA)2(MA)Sn2I7, with achieving a broadband response across an ultraviolet-visible-near infrared (UV-VIS-NIR) region as 365-1064 nm. The high responsivities of 28.4 and 2.0 × 10-2 A/W are obtained at 365 and 1064 nm, respectively, corresponding to detectives of 2.3 × 1010 and 1.8 × 107 Jones. This device also shows remarkable photocurrent stability after 1000 bending cycles. Our work indicates the huge application prospect of Sn-based lead-free perovskites in high-performance and eco-friendly flexible devices.