  We construct a family of four-dimensional noncommutative deformations of
$U(1)$ gauge theory following a general scheme, recently proposed in JHEP 08
(2020) 041 for a class of coordinate-dependent noncommutative algebras. This
class includes the $\mathfrak{su}(2)$, the $\mathfrak{su}(1,1)$ and the angular
(or $\lambda$-Minkowski) noncommutative structures. We find that the presence
of a fourth, commutative coordinate $x^0$ leads to substantial novelties in the
expression for the deformed field strength with respect to the corresponding
three-dimensional case. The constructed field theoretical models are Poisson
gauge theories, which correspond to the semi-classical limit of fully
noncommutative gauge theories. Our expressions for the deformed gauge
transformations, the deformed field strength and the deformed classical action
exhibit flat commutative limits and they are exact in the sense that all orders
in the deformation parameter are present. We review the connection of the
formalism with the $L_{\infty}$ bootstrap and with symplectic embeddings, and
derive the $L_{\infty}$-algebra, which underlies our model.
