We have mapped the starless core TMC-1C in a variety of molecular lines with the IRAM 30 m telescope. High-density tracers show clear signs of self-absorption, and subsonic infall asymmetries are present in N2H+(1-0) and DCO+(2-1) lines. The inward velocity profile in N2H+(1-0) is extended over a region about 7000 AU in radius around the dust continuum peak, which is the most extended ``infalling'' region observed in a starless core with this tracer. The kinetic temperature (~12 K) measured from C17O and C18O suggests that their emission comes from a shell outside the colder interior traced by the millimeter continuum dust. The C18O(2-1) excitation temperature drops from 12 to ~=10 K away from the center. This is consistent with a volume density drop of the gas traced by the C18O lines, from ~=4×104 cm-3 toward the dust peak to ~=6×103 cm-3 at a projected distance from the dust peak of 80" (or 11,000 AU). The column density implied by the gas and dust show similar N2H+ and CO depletion factors (fD<=6). This can be explained with a simple scenario in which: (1) the TMC-1C core is embedded in a relatively dense environment [n(H2)~=104 cm-3], where CO is mostly in the gas phase and the N2H+ abundance had time to reach equilibrium values; (2) the surrounding material (rich in CO and N2H +) is accreting onto the dense core nucleus; (3) TMC-1C is older than 3×105 yr, to account for the observed abundance of N2H + across the core (~=10-10 with respect to H2 ); and (4) the core nucleus is either much younger (~=104 yr) or ``undepleted'' material from the surrounding envelope has fallen toward it in the past 10,000 yr.