  We consider the moduli space $\mathcal{M}_{\nu}$ of torsion-free,
asymptotically conical (AC) Spin(7)-structures which are defined on the same
manifold and asymptotic to the same Spin(7)-cone with decay rate $\nu<0$. We
show that $\mathcal{M}_{\nu}$ is an orbifold if $\nu$ is a generic rate in the
non-$L^2$ regime $(-4,0)$. Infinitesimal deformations are given by topological
data and solutions to a non-elliptic first-order PDE system on the compact link
of the asymptotic cone. As an application, we show that the classical
Bryant-Salamon metric on the bundle of positive spinors on $S^4$ has no
continuous deformations as an AC Spin(7)-metric.
