  We show that $\kappa$-Poincar\'e invariant gauge theories on
$\kappa$-Minkowski space with physically acceptable commutative (low energy)
limit must be 5-d. The gauge invariance requirement of the action fixes the
dimension of the $\kappa$-Minkowski space to $d=5$ and selects the unique
twisted differential calculus with which the construction can be achieved. We
characterize a BRST symmetry related to the 5-d noncommutative gauge invariance
though the definition of a nilpotent operation, which is used to construct a
gauge-fixed action. We also consider standard scenarios assuming
(compactification of) flat extra dimension, for which the 5-d deformation
parameter $\kappa$ can be viewed as the bulk 5-d Planck mass. We study physical
properties of the resulting 4-d effective theories. Recent data from collider
experiments require $\kappa\gtrsim\mathcal{O}(10^{13})\ \text{GeV}$. The use of
standard test of in-vacuo dispersion relations of Gamma Ray Burst photons
increases this lower bound by 4 orders of magnitude. The robustness of this
bound is discussed in the light of possible new features of noncommutative
causal structures.
