As I see it, $p$-adic integers work very similar to formal power series over $x$ (e g. with regards to Hensel lifting).

When it comes to computing $\log P(x)$, one may use the formula

$$ (\log P)' = \frac{P'}{P} $$

to compute the expansion of the logarithm of $P(x)$ with $P(0)=1$ as

$$ \log P \equiv \int \frac{P'}{P} dx \pmod{x^n}. $$

This is the main trick to compute first $n$ coefficients of $\log P(x)$ in $O(M(n))$, where $M(n)$ is the maximum time needed to compute the product of two polynomials of degree at most $n$.

Is there any similar way to compute modulo $p^n$ the expansion of the $p$-adic logarithm of the $p$-adic integer $r$ such that $r \equiv 1 \pmod p$ in $O(M(n))$?

As I see it, the regular notion of polynomial derivatives can't be applied directly to $p$-adic integers, as $(uv)' = u'v+uv'$ woudln't hold. So, maybe there is a way to define some other reasonably invertible function $d(r)$ which is simple enough to compute and such that

$$ d(uv) = u d(v) + v d(u) $$

for any $p$-adic numbers $u$ and $v$?