In this paper we continue to explore the index of elliptic units. In a previous
article we determined the asymptotic behavior in $\boldsymbol{Z}_p$-extensions
of the $p$-part of this index divided by the $p$-part of the ideal class number.
We proved the existence of an invariant $\mu_\infty$ which governs this
behavior, and gave sufficient conditions for the vanishing of $\mu_\infty$. Here
we give examples with nonzero $\mu_\infty$, especially in the case of
anticyclotomic $\boldsymbol{Z}_p$-extensions.
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