  We establish Central Limit Theorems for the volumes of intersections of
$B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of
codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain
higher order approximations for expected volumes, refining previous results by
Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag
version of CLT for convex bodies. We also obtain a Central Limit Theorem for
the Minkowski functional of the intersection body of $B_p^n$, evaluated on a
random vector distributed uniformly on the unit sphere.
