We analyse the cluster discovered by invasion percolation on a branching process with a power-law offspring distribution. Invasion percolation is a paradigm model of self-organised criticality, where criticality is approached without tuning any parameter. By performing invasion percolation for $n$ steps, and letting $n\to\infty$, we find an infinite subtree, called the invasion percolation cluster (IPC). A notable feature of the IPC is its geometry that consists of a unique path to infinity (also called the backbone) onto which finite forests are attached. Our main theorem shows the volume scaling limit of the $k$-cut IPC, which is the cluster containing the root when the edge between the $k$-th and $(k+1)$-st backbone vertices is cut. We assume a power-law offspring distribution with exponent $α$ and analyse the IPC for different power-law regimes. In a finite-variance setting $(α>2)$ our results are a natural extension of previous works on the branching process tree (Michelen et al. 2019) and the regular tree (Angel et al. 2008). However, for an infinite-variance setting ($α\in(1,2)$) or even an infinite-mean setting ($α\in(0,1)$), results significantly change. This is illustrated by the volume scaling of the $k$-cut IPC, which scales as $k^2$ for $α>2$, but as $k^{α/(α-1)}$ for $α\in (1,2)$ and exponentially for $α\in (0,1)$.