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We report on the first observation of stationary light pulses and narrowband light storage inside a hollow-core photonic crystal fiber. Laser-cooled atoms were first loaded into the fiber core providing strong light-matter coupling. Light pulses were then stored in a collective atomic excitation using a single control laser beam. By applying a second counterpropagating control beam, a light pulse could be brought to a standstill. Our work paves the way towards the creation of strongly-correlated many-body systems with photons and applications in the field of quantum information processing.