  Context: In the centre of pre-stellar cores, the deuterium fractionation is
enhanced due to the cold temperatures and high densities. Therefore, the
chemistry of deuterated molecules can be used to probe the evolution and the
kinematics in the earliest stages of star formation. Aims: We analyse emission
maps of cyclopropenylidene, c-C$_3$H$_2$, to study the distribution of the
deuteration throughout the prototypical pre-stellar core L1544. Methods: We use
single-dish observations of c-C$_3$H$_2$, c-H$^{13}$CC$_2$H, c-C$_3$HD, and
c-C$_3$D$_2$ towards the pre-stellar core L1544, performed at the IRAM 30m
telescope. We derive the column density and deuterium fraction maps, and
compare these observations with non-LTE radiative transfer simulations.
Results: The highest deuterium fractions are found close to the dust peak at
the centre of L1544, where the increased abundance of H$_2$D$^+$ ions drives
the deuteration process. The peak values are
N(c-C$_3$HD)/N(c-C$_3$H$_2)=0.17\pm0.01$,
N(c-C$_3$D$_2$)/N(c-C$_3$H$_2)=0.025\pm0.003$ and
N(c-C$_3$D$_2$)/N(c-C$_3$HD$)=0.16\pm0.03$, which is consistent with previous
single point observations. The distributions of c-C$_3$HD and c-C$_3$D$_2$
indicate that the deuterated forms of c-C$_3$H$_2$ in fact trace the dust peak
and not the c-C$_3$H$_2$ peak. Conclusions: The N(c-C$_3$D$_2$)/N(c-C$_3$HD)
map confirms that the process of deuteration is more efficient towards the
centre of the core and demonstrates that carbon-chain molecules are still
present at high densities. This is likely caused by an increased abundance of
He$^+$ ions destroying CO, which increases the amount of carbon atoms in the
gas phase.
