Graphene is being actively explored as a candidate material for flexible and stretchable devices. However, the development of graphene-based flexible photonic devices, i.e. photodetectors, is hindered by the low absorbance of the single layer of carbon atoms. Recently, van der Waals bonded carbon nanotube and graphene hybrid films have demonstrated excellent photoresponsivity, and the use of vein-like carbon nanotube networks resulted in significantly higher mechanical strength. Here, we report for the first time, a flexible photodetector with a high photoresponsivity of ~ 51 A/W and a fast response time of ~ 40 ms over the visible range, revealing the unique potential of this emerging all-carbon hybrid films for flexible devices. In addition, the device exhibits good robustness against repetitive bending, suggesting its applicability in large-area matrix-array flexible photodetectors.