A promising approach to merge atomic systems with scalable photonics has emerged recently, which consists of trapping cold atoms near tapered nanofibers. Here, we describe a novel technique to achieve strong, coherent coupling between a single atom and photon in such a system. Our approach makes use of collective enhancement effects, which allow a lattice of atoms to form a high-finesse cavity within the fiber. We show that a specially designated ‘impurity’ atom within the cavity can experience strongly enhanced interactions with single photons in the fiber. Under realistic conditions, a ‘strong coupling’ regime can be reached, wherein it becomes feasible to observe vacuum Rabi oscillations between the excited impurity atom and a single cavity quantum. This technique can form the basis for a scalable quantum information network using atom-nanofiber systems.