  We consider a non-Abelian Lee--Wick gauge theory and discuss
Becchi-Rouet-Stora-Tyutin (BRST) invariance. It contains fourth-order
derivative as extensions of the kinetic term, leading to massive ghosts in the
theory upon quantization. We particularly provide essential clues towards
confinement conditions in strongly-coupled regimes, using the Kugo-Ojima
approach, and obtain the $\beta-$functions in the non-perturbative regimes.
This is achieved using a set of exact solutions of the corresponding local
theory in terms of Jacobi elliptical functions. We obtain a similar
$\beta-$function just as for the ordinary Yang-Mills theory but the main
differences are that now, the cut-off arises naturally from the Lee-Wick heavy
mass scales (M). We show that the fate of the ghosts is fixed in these regimes:
they are no more the propagating degrees of freedom in the infrared (IR)-limit.
As it also happens for the ordinary case, confinement is due to the non-Abelian
nature of the theory. In the limit $M\rightarrow\infty $, one recovers the
standard results for the local non-Abelian Yang-Mills theory.
