  We report on the low-energy dynamics in the kagome antiferromagnet
CaCu$_3$(OD)$_6$Cl$_2\cdot0.6$D$_2$O (Ca-kapellasite) as studied by use of
$^2$D-NMR measurements. Previous $^{35}$Cl-NMR measurements revealed that the
nuclear spin-lattice relaxation rate ($1/T_1$) shows two peaks at temperatures,
$T^{\ast} = 7.2$ K and $T_s \simeq 25$ K. While the low-temperature peak at
$T^{\ast}$ is ascribed to the critical fluctuations near the long-range
magnetic ordering, the origin of the high-temperature peak has not been fully
understood. From the $1/T_1$ measurements on the D sites at the OD groups
(D$_{\rm OD}$), we find no peak at $T_s$, evidencing that the high-temperature
peak is not related to the molecular dynamics of the OD groups. We discuss the
possibility of a frustration-induced short-range ordered state below $T_s$
before the long-range order is stabilized by the Dzyaloshinskii-Moriya
interaction. We also observed static internal fields at the D$_{\rm OD}$ site
in the long-range ordered state below $T^{\ast}$, and confirm the previously
proposed negative-chirality $q=0$ magnetic structure.
