  We present a survey for neutral atomic-carbon (CI) along gamma-ray burst
(GRB) sightlines, which probes the shielded neutral gas-phase in the
interstellar medium (ISM) of GRB host galaxies at high redshift. We compile a
sample of 29 medium- to high-resolution GRB optical afterglow spectra spanning
a redshift range through most of cosmic time from $1 < z < 6$. We find that
seven ($\approx 25\%$) of the GRBs entering our statistical sample have CI
detected in absorption. It is evident that there is a strong excess of cold gas
in GRB hosts compared to absorbers in quasar sightlines. We investigate the
dust properties of the GRB CI absorbers and find that the amount of neutral
carbon is positively correlated with the visual extinction, $A_V$, and the
strength of the 2175 \AA\ dust extinction feature, $A_{\mathrm{bump}}$. GRBs
with CI detected in absorption are all observed above a certain threshold of
$\log N$(HI)$/\mathrm{cm}^{-2}$ + [X/H] > 20.7 and a dust-phase iron column
density of $\log N$(Fe)$_{\mathrm{dust}}/\mathrm{cm}^{-2}$ > 16.2. In contrast
to the SED-derived dust properties, the strength of the CI absorption does not
correlate with the depletion-derived dust properties. This indicates that the
GRB CI absorbers trace dusty systems where the dust composition is dominated by
carbon-rich dust grains. The observed higher metal and dust column densities of
the GRB CI absorbers compared to H$_2$- and CI-bearing quasar absorbers is
mainly a consequence of how the two absorber populations are selected, but is
also required in the presence of intense UV radiation fields in actively
star-forming galaxies.
