By a Banach space $X$ I mean, a complete normed vector space and by a Clifford isometry I mean a surjective isometry $\gamma$ of $X$ such that the distance $d(\gamma x, x)$ is constant on $X$. Inherently, $\gamma$ acts as a "translation" so Clifford isometries are sometimes called Clifford translations. As an example, in Euclidean space any translation "is" a Clifford isometry.
My question is: are there examples of such an $X$ where the set of Clifford isometries consists only of the identity?