We consider a boundary value problem for the $p$-Laplacian, posed in the exterior of small cavities that all have the same $p$-capacity and are anchored to the unit sphere in $\mathbb{R}^d$, where $1<p<d.$ We assume that the distance between anchoring points is at least $\varepsilon$ and the characteristic diameter of cavities is $α\varepsilon$, where $α=α(\varepsilon)$ tends to 0 with $\varepsilon$. We also assume that anchoring points are asymptotically uniformly distributed as $\varepsilon \downarrow 0$, and their number is asymptotic to a positive constant times $\varepsilon^{1-d}$. The solution $u=u^\varepsilon$ is required to be 1 on all cavities and decay to 0 at infinity. Our goal is to describe the behavior of solutions for small $\varepsilon>0$. We show that the problem possesses a critical window characterized by $τ:=\lim_{\varepsilon \downarrow 0}α/α_c \in (0,\infty)$, where $α_c=\varepsilon^{1/γ}$ and $γ= \frac{d-p}{p-1}.$ We prove that outside the unit sphere, as $\varepsilon\downarrow 0$, the solution converges to $A_*U$ for some constant $A_*$, where $U(x)=\min\{1,|x|^{-γ}\}$ is the radial $p$-harmonic function outside the unit ball. Here the constant $A_*$ equals 0 if $τ=0$, while $A_*=1$ if $τ=\infty$. In the critical window where $τ$ is positive and finite, $ A_*\in(0,1)$ is explicitly computed in terms of the parameters of the problem. We also evaluate the limiting $p$-capacity in all three cases mentioned above. Our key new tool is the construction of an explicit ansatz function $u_{A_*}^\varepsilon$ that approximates the solution $u^\varepsilon$ in $L^{\infty}(\mathbb{R}^d)$ and satisfies $\|\nabla u^\varepsilon-\nabla u_{A_*}^\varepsilon \|_{L^{p}(\mathbb{R}^d)} \to 0$ as $\varepsilon \downarrow 0$.