We consider a branching Brownian motion in $\mathbb R^d$ with $d \geq 1$ in which the position $X_t^{(u)}\in \mathbb R^d$ of a particle $u$ at time $t$ can be encoded by its direction $\theta^{(u)}_t \in \mathbb S^{d-1}$ and its distance $R^{(u)}_t$ to 0. We prove that the \emph{extremal point process} $\sum \delta_{\theta^{(u)}_t, R^{(u)}_t - m_t^{(d)}}$ (where the sum is over all particles alive at time $t$ and $m^{(d)}_t$ is an explicit centring term) converges in distribution to a randomly shifted decorated Poisson point process on $\mathbb S^{d-1} \times \mathbb R$. More precisely, the so-called {\it clan-leaders} form a Cox process with intensity proportional to $D_\infty(\theta) e^{-\sqrt{2}r} \mathrm d r\mathrm d \theta $, where $D_\infty(\theta)$ is the limit of the derivative martingale in direction $\theta$ and the decorations are i.i.d.\ copies of the decoration process of the standard one-dimensional branching Brownian motion. This proves a conjecture of Stasi\'nski, Berestycki and Mallein (Ann.\ Inst.\ H.\ Poincar\'{e} 57:1786--1810, 2021), and builds on that paper and on Kim, Lubetzky and Zeitouni (arXiv:2104.07698).