= Solution
Choose an integral <Minimal Weierstrass equation> over $\mathbb Z_p$. Let $E_0(\mathbb Q_p)$ be the points with nonsingular reduction, and let $E_1(\mathbb Q_p)$ be those reducing to $O$. This distinction matters at <bad reduction of an elliptic curve>: the whole singular reduced cubic is not a group. The smooth locus is a group under the induced <elliptic curve group law>, and <Hensel's lemma> lifts each of its <rational points>, giving
$$
0\longrightarrow E_1(\mathbb Q_p)\longrightarrow E_0(\mathbb Q_p)\longrightarrow\widetilde E_{\mathrm{ns}}(\mathbb F_p)\longrightarrow0.
$$
The <formal kernel of a minimal Weierstrass equation> identifies $E_1(\mathbb Q_p)$ with $p\mathbb Z_p$ equipped with $F$. Indeed, the integral series $w(t)$ converges there, and $x=t/w(t)$, $y=-1/w(t)$ give the unique point for each nonzero parameter. If $v_p(t)=r>0$, then $v_p(x)=-2r$ and $v_p(y)=-3r$; the parameter zero gives $O$. Conversely a point reducing to $O$ has these local coordinates of positive <valuation>, so is recovered by the series.
Define $E_m$ for $m\ge1$ by $t\in p^m\mathbb Z_p$. The linear part of the <formal group law> is addition, and the higher terms have degree at least two. Hence the <filtration of elliptic-curve points over a local field> has
$$
E_m/E_{m+1}\cong(p^m\mathbb Z_p/p^{m+1}\mathbb Z_p,+)\cong(\mathbb F_p,+).
$$
At <good reduction>, $E_0=E$ and $E/E_1\cong\widetilde E(\mathbb F_p)$. At <bad reduction>, there is also the finite quotient $E/E_0$: $E(\mathbb Q_p)$ is a <compact group> because the curve is projective, and $E_0$ is an open subgroup, so its <index of a subgroup> is finite. Together these quotients and the formal filtration describe the group, including the extra components at bad primes.
The <formal logarithm> makes the deepest part explicit. Write $L(T)=T+\sum_{n\ge2}c_nT^n$. Since its derivative is the integral <invariant differential of a formal group law>, $nc_n\in\mathbb Z_p$. Consequently
$$
v_p(c_nt^n)\ge n\,v_p(t)-v_p(n).
$$
The inequality $v_p(n)\le(n-1)/(p-1)$ shows that for an integer $m>1/(p-1)$ every nonlinear term on $p^m\mathbb Z_p$ has strictly higher <valuation> than the linear term. After scaling, $L$ is the identity plus a strictly contracting series; solving $L(t)=u$ by successive correction gives its inverse <formal group exponential>. Thus the <deep logarithm subgroup of a formal group> gives
$$
\boxed{E_m(\mathbb Q_p)\cong(p^m\mathbb Z_p,+)\quad\text{when }m>1/(p-1).}
$$
One may take $m=1$ for odd $p$ and $m=2$ for $p=2$. In particular, $E(\mathbb Q_p)$ contains an open subgroup isomorphic to the additive <p-adic integers>, and the remaining information is in finite quotients and their group extensions.
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