Carbon abundances in late-type stars are important in a variety of astrophysical contexts. However C I lines, one of the main abundance diagnostics, are sensitive to departures from local thermodynamic equilibrium (LTE). We present a model atom for non-LTE analyses of C I lines, that uses a new, physically-motivated recipe for the rates of neutral hydrogen impact excitation. We analyse C I lines in the solar spectrum, employing a three-dimensional (3D) hydrodynamic model solar atmosphere and 3D non-LTE radiative transfer. We find negative non-LTE abundance corrections for C I lines in the solar photosphere, in accordance with previous studies, reaching up to around 0.1 dex in the disk-integrated flux. We also present the first fully consistent 3D non-LTE solar carbon abundance determination: we infer log is an element of(C) = 8.44 ± 0.02, in good agreement with the current standard value. Our models reproduce the observed solar centre-to-limb variations of various C I lines, without any adjustments to the rates of neutral hydrogen impact excitation, suggesting that the proposed recipe may be a solution to the long-standing problem of how to reliably model inelastic collisions with neutral hydrogen in late-type stellar atmospheres.