  In this paper, we are interested in the self-similar growth-fragmentation
process that shows up when slicing half-space excursions of a $d$-dimensional
Brownian motion from hyperplanes. Such a family of processes turns out to be a
spatial self-similar growth-fragmentation processes driven by an isotropic
self-similar Markov process. The former can be seen as multitype
growth-fragmentation processes, in the sense of arXiv:2112.11091, where the set
of types is $\mathbb{S}^{d-2}$, the $(d-1)$-dimensional unit sphere. In order
to characterise such family of processes, we study their spinal description
similarly as in the monotype and multitype settings. Finally, we extend our
study to the case when the $d$-dimensional Brownian motion is replaced by an
isotropic Markov process whose first $(d-1)$ coordinates are driven by an
isotropic stable L\'evy process and the remaining coordinate is an independent
standard real-valued Brownian motion.
