  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 which enforces spatial coherence when fitting multiple velocity
components simultaneously over a data cube. We analyze HC$_5$N (9-8) line
emission as part of the Green Bank Ammonia Survey (GAS) 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 $2.7
\pm 0.1$ 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
HC$_5$N 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 HC$_5$N 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.
