 Solution

Reduce $P$ to $\overline{P}\in \tilde{E}({\mathbb{F}}_p)$. Because $p\nmid n$, the multiplication-by-n morphism on the reduced elliptic curve is a separable, surjective morphism of degree $n^2$. Choose a point $\overline{Q}\in \tilde{E}({\overline{\mathbb{F}}}_p)$ with $n\overline{Q}=\overline{P}$. Its coordinates lie in some finite field ${\mathbb{F}}_{p^d}$.

Let $L/{\mathbb{Q}}_p$ be the finite unramified extension with residue field ${\mathbb{F}}_{p^d}$. The curve retains good reduction over $L$. Its smooth integral model and the Hensel lemma give surjectivity of good reduction over a local field, so lift $\overline{Q}$ to some $Q_0\in E(L)$. This lift need not satisfy $n{Q}_0=P$ exactly, but its errorhas zero reduction and belongs to $E_1(L)$, the kernel of reduction of an elliptic curve.

The parameter $t=-x/y$ at infinity identifies $E_1(L)$ with the maximal ideal ${\mathfrak{m}}_L$ equipped with the formal group of an elliptic curve. The integral power series for $[n]$ and its inverse from part (i) remain integral over ${\mathcal{O}}_L$. They converge on ${\mathfrak{m}}_L$: terms of degree $k$ tend to zero in valuation because their coefficients are integral and the argument has positive valuation. Hence multiplication by $n$ is bijective on $E_1(L)$. Choose its unique solution $T\in E_1(L)$ to $nT=R$, and put $Q=Q_0+T$. ThenThe residue-field extension supplies a division point modulo $p$, and the formal-group inverse corrects the error without introducing ramification. This is the local construction underlying the uniform unramified division field over a local field and unramified division torsors at good primes.