  Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic
manifold with a conformal infinity $(M^n, [\hat{h}])$. The fractional Yamabe
problem addresses to solve \[P^{\gamma}[g^+,\hat{h}] (u) = cu^{n+2\gamma \over
n-2\gamma}, \quad u > 0 \quad \text{on } M\] where $c \in \mathbb{R}$ and
$P^{\gamma}[g^+,\hat{h}]$ is the fractional conformal Laplacian whose principal
symbol is $(-\Delta)^{\gamma}$. In this paper, we construct a metric on the
half space $X = \mathbb{R}^{n+1}_+$, which is conformally equivalent to the
unit ball, for which the solution set of the fractional Yamabe equation is
non-compact provided that $n \ge 24$ for $\gamma \in (0, \gamma^*)$ and $n \ge
25$ for $\gamma \in [\gamma^*,1)$ where $\gamma^* \in (0, 1)$ is a certain
transition exponent. The value of $\gamma^*$ turns out to be approximately
0.940197.
