Let $p$ be an odd prime number, $m$ a positive integer with $p\mid m$. Put $k=\mathbf{Q}(\mu_m)$.

(1) Is there any example where certain *noncyclotomic* $\mathbf{Z}_p$-extension $k_\infty/k$ has positive $\mu$-invariant?

(2) Let $n$ be an integer such that $m\mid n$ and $p$ does not divide $n/m$. Put $k'=\mathbf{Q}(\mu_n)$, $k'_\infty=k_\infty\cdot k'$. It is well-known that $\mu(k_\infty/k)\le \mu(k'_\infty/k')$. I wonder is it possible that the two sides differ by exactly 1. Here, for this part, I allow $\mu(k_\infty/k)=0$. I know that Iwasawa and Ozaki has constructed examples from cyclotomic fields which have positive $\mu$-invariant. But their examples all have degree divisible by $p$ over the base field. Their method does not apply to the above question.