How rigid the finite ultrametric spaces can be?

A metric space X is rigid if the isometry group of X is trivial. The finite ultrametric spaces X with |X| ≥ 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.