= Solution
Put $I=\{x:|C(x)|=\infty\}$, the union of all infinite <percolation clusters>. Since $p>p_c$, the <percolation probability> $\theta(p)$ is positive. The <translation ergodicity of Bernoulli percolation> makes $\{I\ne\varnothing\}$ a zero-one event; it has positive probability because $\mathbb P_p(0\in I)=\theta(p)>0$, hence it occurs <almost surely>.
Let $D_m=\{B(m)\cap I\ne\varnothing\}$. These events increase to $\{I\ne\varnothing\}$, so <continuity from below of a measure> gives $\mathbb P_p(D_m)\to1$. On $D_m$, some vertex of $B(m)$ has an infinite open <path in a graph>. For every $N\geq m$, the segment up to its first visit to $\partial B(N)$ lies entirely in $B(N)$. Thus even with this restriction on the connecting <path in a graph>,
$$
\boxed{\inf_{n\geq1}\mathbb P_p\bigl(B(m)\longleftrightarrow\partial B(m+n)\text{ within }B(m+n)\bigr)\geq\mathbb P_p(D_m)\longrightarrow1.}
$$
This proves the requested uniformity and supplies the version needed for crossings later.
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