  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 {\it 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).
