about the p-adic logarithm It is well known that we can define the logarithm in the p-adic setup by the usual power series that converges in $B(1,1^-)$. In Schickoff "Ultrametric calculus" there is an extension of the $log$ from the unit ball to $C_p^{\times}$ (called $LOG$) and it is proved that it is locally analytic, then he defines the Iwasawa Logarithm as the function $LOG(x) -ord_p(x)$ and this is the unique multiplicative function extending the logarithm of the unit ball such that vanishes on $p$. My question is, this function is still locally analytic on $C_p^{\times}$? Thanks for the answers.

Sure. Local analyticity can be judged solely from the behavior at the identity, $1$. And there, the function is given by the series that you know.