Filamentary structures have been found nearly ubiquitously in molecular clouds and yet their formation and evolution is still poorly understood. We examine a segment of Taurus Molecular Cloud 1 (TMC-1) that appears as a single, narrow filament in continuum emission from dust. We use the Regularized Optimization for Hyper-Spectral Analysis (ROHSA), a Gaussian decomposition algorithm that enforces spatial coherence when fitting multiple velocity components simultaneously over a data cube. We analyse HC5N (9-8) line emission as part of the Green Bank Ammonia Survey and identify three velocity-coherent components with ROHSA. The two brightest components extend the length of the filament, while the third component is fainter and clumpier. The brightest component has a prominent transverse velocity gradient of 2.7 +/- 0.1 km s(-1) pc(-1) that we show to be indicative of gravitationally induced inflow. In the second component, we identify regularly spaced emission peaks along its length. We show that the local minima between pairs of adjacent HC5N peaks line up closely with submillimetre continuum emission peaks, which we argue is evidence for fragmentation along the spine of TMC-1. While coherent velocity components have been described as separate physical structures in other star-forming filaments, we argue that the two bright components identified in HC5N emission in TMC-1 are tracing two layers in one filament: a lower density outer layer whose material is flowing under gravity towards the higher density inner layer of the filament.