Let (𝑋, 𝑑) be an ultrametric space and let 𝑑𝐻 be the Hausdorff distance on the set B¯ 𝑋 of all closed balls in (𝑋, 𝑑). Some interconnections between the properties of the spaces (𝑋, 𝑑) and (B¯ 𝑋, 𝑑𝐻 ) are described. It is established that the space (B¯ 𝑋, 𝑑𝐻 ) has such properties as discreteness, local finiteness, metrical discreteness, completeness, compactness, local compactness if and only if the space (𝑋, 𝑑) has these properties. Necessary and sufficient conditions for the separability of the space (B¯ 𝑋, 𝑑𝐻 ) are also proved.