= Solution
Because $p<p_c=\widetilde p_c$, choose a finite set $S\ni0$ with $\phi_p(S)=\rho<1$. Let $L$ exceed the $\ell^\infty$-distance from $0$ to every endpoint of an edge in $\partial S$, and put
$$
q_n=\mathbb P_p(0\leftrightarrow\partial B(n)).
$$
On the one-arm event to distance $n>L$, take the first oriented boundary edge $(x,y)\in\partial S$ used by an open self-avoiding path. The connection $0\xleftarrow{S}x$, the open edge $(x,y)$, and the remaining connection from $y$ to $\partial B(n)$ occur disjointly. The <van den Berg-Kesten inequality> and translation invariance therefore give
$$
\begin{aligned}
q_n
&\leq p\sum_{(x,y)\in\partial S}
\mathbb P_p(0\xleftarrow{S}x)
\mathbb P_p(y\leftrightarrow\partial B(n))\\
&\leq\phi_p(S)q_{n-L}=\rho q_{n-L}.
\end{aligned}
$$
Iteration yields $q_n\leq\rho^{\lfloor n/L\rfloor}$ up to an inessential finite-scale adjustment. Since $p<1$, each of the finitely many remaining $q_n$ is strictly below one, so reducing the exponent if necessary produces a constant $c>0$ valid for every $n\geq1$:
$$
\boxed{\mathbb P_p(0\leftrightarrow\partial B(n))\leq e^{-cn}.}
$$
This is the finite-size proof of <exponential decay of subcritical percolation>.
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