We establish that if $d \geq 2k + 6$ and $q$ is odd and sufficiently large with respect to $α\in (0,1)$, then every set $A\subseteq \mathbf{F}_q^d$ of size $|A| \geq αq^d$ will contain an isometric copy of every spherical $(k+2)$-point configuration that spans $k$ dimensions.