  A metric space $X$ is rigid if the isometry group of $X$ is trivial. The
finite ultrametric spaces $X$ with $|X| \geq 2$ are not rigid since for every
such $X$ there is a self-isometry having exactly $|X|-2$ fixed points. Using
the representing trees we characterize the finite ultrametric spaces $X$ for
which every self-isometry has at least $|X|-2$ fixed points. Some other
extremal properties of such spaces and related graph theoretical
characterizations are also obtained.
