  In this paper, we study the influence of spatial fluctuations in a
two-dimentional Kondo-Lattice model (KLM) with anti-ferromagnetic couplings. To
accomplish this, we first present an implementation of the dual-fermion (DF)
approach based on the hybridization expansion continuous-time quantum Monte
Carlo impurity solver (CT-HYB), which allows us to consistently compare the
local and non-local descriptions of this model. We find that, the inclusion of
non-locality restores the self-energy dispersion of the conduction electrons,
{\it i.e.} the $\vec{k}$ dependence of $\Sigma(\vec{k}, i\omega_{n})$. The
anti-ferromagnetic correlations result in an additional symmetry in
$\Sigma(\vec{k}, i\omega_{n})$, which is well described by the N\'eel
antiferromagnetic wave-vector. A "metal"-"anti-ferromagnetic insulator"-"Kondo
insulator" transition is observed at finite temperatures, which is driven by
the competition of the effective RKKY interaction (at the weak coupling regime)
and the Kondo singlet formation mechanism (at the strong coupling regime). Away
from half-filling, the anti-ferromagnetic phase becomes unstable against hole
doping. The system tends to develop a ferromagnetic phase with the spin
susceptibility $\chi_{s}(Q)$ peaking at $Q=\Gamma$. However, for small $J/t$,
no divergence of $\chi_{s}(\Gamma)$ is really observed, thus, we find no sign
of long-range ferromagnetism in the hole-doped two-dimension KLM. The
ferromagnetism is found to be stable at larger $J/t$ regime. Interestingly, we
find the local approximation employed in this work, {\it i.e.} the dynamical
mean-field theory (DMFT), is still a very good description of the KLM,
especially in the hole-doped case. However, at half-filling, the non-local
fluctuation effect is indeed pronounced. We observe a strong reduction of the
critical coupling strength for the onset of the Kondo insulating phase.
