= Solution
The objective is a continuous <faithful representation> of a <uniform pro-p group> $G$ in some $\operatorname{GL}_N(\mathbb Z_p)$. The proof proceeds from the intrinsic Lie lattice to a small <matrix group>, then extends the resulting representation to all of $G$. This last step matters: a representation defined only on an <open subgroup> does not yet establish linearity of the original <group>.
Let $L=L_G$ be the <Lie algebra of a uniform pro-p group> and put $\mathfrak g=\mathbb Q_p\otimes_{\mathbb Z_p}L$. The uniform-group/Lie-lattice correspondence identifies the <group> law on $L$ with BCH. The substantial finite-dimensional Lie-theoretic input is the <Ado theorem>: every finite-dimensional <Lie algebra> over a characteristic-zero <field> admits a faithful finite-dimensional representation. Apply it to obtain
$$
\rho:\mathfrak g\hookrightarrow\operatorname{End}_{\mathbb Q_p}(V),\qquad\rho([x,y])=\rho(x)\rho(y)-\rho(y)\rho(x).
$$
An <Adjoint representation> would not suffice here, since it kills the centre; Ado supplies <injectivity> even in central directions.
Choose a <p-adic lattice> $\Lambda$ spanning $V$. Since $L$ has finitely many lattice generators, their representing <matrices> have bounded denominators. There is therefore an integer $a\ge0$ such that
$$
\rho(p^aL)\subseteq p^\epsilon\operatorname{End}_{\mathbb Z_p}(\Lambda),\qquad\epsilon=\begin{cases}1,&p\ne2,\\2,&p=2.\end{cases}
$$
On this <matrix> lattice the series $\exp X=\sum_{n\ge0}X^n/n!$ and $\log(1+Y)=\sum_{n\ge1}(-1)^{n+1}Y^n/n$ converge and are inverse. For example $v_p(n!)\le n/(p-1)$, so the valuations of the exponential terms tend to infinity with this choice of $\epsilon$; the <logarithm> terms do as well. They take values in $1+p^\epsilon\operatorname{End}(\Lambda)$ and in $p^\epsilon\operatorname{End}(\Lambda)$ respectively. The formal identity $\exp(\operatorname{BCH}(X,Y))=\exp X\exp Y$ holds for the convergent series.
Under the correspondence, $U=G^{p^a}$ has Lie lattice $p^aL$ and is an open <characteristic subgroup>. Define
$$
\eta:U\longrightarrow\operatorname{GL}(\Lambda),\qquad\eta(u)=\exp(\rho(\log_Gu)),
$$
where $\log_G$ denotes the identification of $U$ with $p^aL$. A Lie <homomorphism> preserves BCH, so the exponential identity proves $\eta(uv)=\eta(u)\eta(v)$. If $\eta(u)=I$, the <matrix> <logarithm> gives $\rho(\log_Gu)=0$; faithfulness of $\rho$ gives $u=1$. Thus $\eta$ is a continuous faithful integral representation of $U$.
To extend it, take the <module>
$$
W=\{f:G\to\Lambda:f(ux)=\eta(u)f(x)\text{ for all }u\in U,\ x\in G\}.
$$
Values on a transversal of $U\backslash G$ determine $f$, so $W$ is a <free module> over $\mathbb Z_p$ of rank $[G:U]\dim V$. These functions are continuous, since on each open <coset> they are given by the continuous representation $\eta$. Define $(T_gf)(x)=f(xg)$. The equivariance condition is preserved, and $T_gT_h=T_{gh}$. In a transversal <basis> each $T_g$ is a permutation of blocks followed by <matrices> from $\eta(U)$, so it belongs to $\operatorname{GL}(W)$ and depends continuously on $g$.
Faithfulness can be checked directly. If $g\notin U$, right multiplication moves the <coset> $U$ to a different <coset>, so $T_g$ moves a function supported on one block and cannot be the identity. If $g\in U$ and $T_g=1$, evaluation at $1$ gives $f(g)=f(1)$ for every $f$, whereas equivariance gives $f(g)=\eta(g)f(1)$. Since $f(1)$ is arbitrary, $\eta(g)=I$, and hence $g=1$. Therefore
$$
\boxed{G\hookrightarrow\operatorname{GL}_{[G:U]\dim V}(\mathbb Z_p).}
$$
<Compactness> of $G$ makes this continuous injection a <homeomorphism> onto its closed image. This proves the <linearity of a uniform pro-p group> for the full <group>, including its centre, and explains why taking a sufficiently small power lattice permits <matrix> integration before finite-index induction restores the whole <group>.
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