  We consider the fuzzy 4-sphere $S_N^4$ as a background in the IKKT matrix
model and explore the relation between $S_N^4$ and fuzzy twistor space in the
semi-classical limit. A novel description for the IKKT-matrix model in terms of
spinorial indices is given, which is reminiscent of $\mathcal{N}=4$
super-symmetric Yang-Mills (SYM) in $4d$. On fuzzy twistor space, the
interactions of the IKKT model are of gravitational type. The higher-spin (HS)
gauge theory emerging in this limit from the IKKT model, denoted as HS-IKKT, on
fuzzy twistor space is shown to be a higher-spin extension of $\mathcal{N}=4$
SYM, with vertices that have more than two derivatives. We obtain its
(Euclidean) spacetime action using the Penrose transform. Although this is a
gravitational theory, it shares many features with the higher-spin extensions
of Yang-Mills in $4d$ flat space obtained in arXiv:2105.12782,
arXiv:2107.04500. The tree-level amplitudes of the HS-IKKT are studied in the
semi-classical flat limit. The self-dual sector of the IKKT model is obtained
by dropping some parts of the cubic- and the quartic interactions, which is
shown to reduce to a BF-type action on commutative deformed projective twistor
space.
