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If $(X,d)$ is a metric space and $\epsilon \in (0,1)$, then $(X, d^\epsilon)$ is also a metric space (called a ``snowflake'' of $(X,d)$). It seems to me that if the former space is nice, then so is the latter, trivially. The snowflake will not be injective. For a concrete example, take $X=[0,1]$ and $\epsilon=\frac{1}{2}$. Does this work or have I made a mistake?
