We report on the low-energy dynamics in the kagome antiferromagnet CaCu$_3$(OD)$_6$Cl$_2cdot0.6$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.
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