  We introduce the notion of (almost isometric) local retracts in metric space
as a natural non-linear version of the concepts of locally complemented and
almost isometric ideals from Banach spaces. We prove that given two metric
spaces $N\subseteq M$ there always exists an almost isometric local retract
$S\subseteq M$ with $N\subseteq S$ and $dens(N)=dens(S)$. We also prove that
metric spaces which are local retracts (respectively almost isometric local
retracts) can be characterised in terms of a condition of extendability of
Lipschitz functions (respectively almost isometries) between finite metric
spaces. Different examples and counterexamples are exhibited.
