  We derive a sharp upper bound for the first eigenvalue $\lambda_{1,p}$ of the
$p$-Laplacian on asymptotically hyperbolic manifolds for $1<p<\infty$. We then
prove that a particular class of conformally compact submanifolds within
asymptotically hyperbolic manifolds are themselves asymptotically hyperbolic.
As a corollary, we show that for any minimal conformally compact submanifold
$Y^{k+1}$ within $\mathbb{H}^{n+1}(-1)$,
$\lambda_{1,p}(Y)=\left(\frac{k}{p}\right)^{p}$. We then obtain lower bounds on
$\lambda_{1,2}(Y)$ in the case where minimality is replaced with a bounded mean
curvature assumption and where the ambient space is a general
Poincar\'e-Einstein space whose conformal infinity is of non-negative Yamabe
type. In the process, we introduce an invariant $\hat \beta^Y$ for each such
submanifold, enabling us to generalize a result due to Cheung-Leung.
