= Solution
Near $O$, put $t=-x/y$ and $w=-1/y$, or projectively $t=-X/Y$, $w=-Z/Y$. The integral <Weierstrass equation of an elliptic curve> becomes
$$
w=t^3+a_1tw+a_2t^2w+a_3w^2+a_4tw^2+a_6w^3.
$$
The linear coefficient of $w$ at the origin is a unit, so successive coefficient comparison gives a unique series $w(t)\in\mathbb Z_p[[t]]$ beginning $t^3$. Expressing the addition morphism in the local parameters of its two inputs gives
$$
F(t_1,t_2)=t_1+t_2+\text{terms of total degree at least two}\in\mathbb Z_p[[t_1,t_2]].
$$
Its identity, inverse, commutativity and associativity identities follow from the corresponding elliptic <group> identities. This one-dimensional commutative <formal group law> is the <formal group of an elliptic curve> $\widehat E$ attached to the model.
A point reducing to $O$ lies in the chart with $Y$ a unit and has $t,w\in p\mathbb Z_p$. The implicit equation determines $w$ uniquely from $t$. Conversely every $t\in p\mathbb Z_p$ gives the convergent solution $w(t)$ and the point $[-t:1:-w(t)]$; $t=0$ gives $O$. Hence the parameter is a bijection from the reduction kernel to $p\mathbb Z_p$. The series $F$ converges there and describes its addition. Therefore
$$
\boxed{\ker\varphi=E_1(\mathbb Q_p)\cong\widehat E(p\mathbb Z_p).}
$$
For nonzero $t$ with $v_p(t)=r>0$, $w(t)=t^3$ times a unit, so $v_p(x)=-2r$ and $v_p(y)=-3r$. These are the <formal kernel of a minimal Weierstrass equation> <valuations> used earlier.
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