We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $ρ$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $ρ$. We prove that for every dimension $d$ and positive reals $ρ$ and $α$ there exists a positive $β=β(d,ρ, α)$ such that if $\mathcal{F}$ is a finite family of $ρ$-fat convex sets in $\mathbb{R}^d$ and an $α$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $β$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls.