  As an extension of the weak perturbation theory obtained in recent analysis
on infinite-derivative non-local non-Abelian gauge theories motivated from
p-adic string field theory, and postulated as direction of UV-completion in 4-D
Quantum Field Theory (QFT), here we investigate the confinement conditions and
$\beta-$function in the strong coupling regime. We extend the confinement
criterion, previously obtained by Kugo and Ojima for the local theory based on
the Becchi-Rouet-Stora-Tyutin (BRST) invariance, to the non-local theory, by
using a set of exact solutions of the corresponding local theory. We show that
the infinite-derivatives which are active in the UV provides finite
contributions also in the infrared (IR) limit and provide a proof of
confinement, granted by the absence of the Landau pole. The main difference
with the local case is that the IR fixed point is moved to infinity. We also
show that in the limit of the energy scale of non-locality $M \rightarrow
\infty$ we reproduce the local theory results and see how asymptotic freedom is
properly recovered.
