= Solution
Reduce $P$ to $\overline P\in\widetilde E(\mathbb F_p)$. 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 $\overline Q\in\widetilde E(\overline{\mathbb F}_p)$ 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 $nQ_0=P$ exactly, but its error
$$
R=P-nQ_0
$$
has 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 $Q=Q_0+T$. Then
$$
\boxed{Q\in E(L),\qquad nQ=nQ_0+R=P,\qquad L/\mathbb Q_p\text{ finite unramified}.}
$$
The 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>.
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