Say $K$ is a complete nonarchimedean extension of $\mathbf{Q}_p$, i.e., it is the fraction field of a $p$-adically complete and $p$-torsionfree rank $1$ valuation ring. Assume that the residue field of $K$ is perfect, and that one of the following two conditions holds:

a) $\mu_{p^\infty} \in K$.

b) $p$ admits a compatible system of $p$-power
roots in $K$.

Is $K$ perfectoid?