  We construct a non-commutative kappa-Minkowski deformation of U(1) gauge
theory, following a general approach, recently proposed in JHEP 2008 (2020)
041. We obtain an exact (all orders in the non-commutativity parameter)
expression for both the deformed gauge transformations and the deformed field
strength, which is covariant under these transformations. The corresponding
Yang-Mills Lagrangian is gauge covariant and reproduces the Maxwell Lagrangian
in the commutative limit. Gauge invariance of the action functional requires a
non-trivial integration measure which, in the commutative limit, does not
reduce to the trivial one. We discuss the physical meaning of such a nontrivial
commutative limit, relating it to a nontrivial space-time curvature of the
undeformed theory. Moreover, we propose a rescaled kappa-Minkowski
non-commutative structure, which exhibits a standard flat commutative limit.
