  We consider the discrete Gaussian Free Field in a square box in $\mathbb Z^2$
of side length $N$ with zero boundary conditions and study the joint law of its
properly-centered extreme values ($h$) and their scaled spatial positions ($x$)
in the limit as $N\to\infty$. Restricting attention to extreme local maxima,
i.e., the extreme points that are maximal in an $r_N$-neighborhood thereof, we
prove that the associated process tends, whenever $r_N\to\infty$ and
$r_N/N\to0$, to a Poisson point process with intensity measure $Z(dx)e^{-\alpha
h}dh$, where $\alpha:= 2/\sqrt{g}$ with $g:=2/\pi$ and where $Z(dx)$ is a
random Borel measure on $[0,1]^2$. In particular, this yields an integral
representation of the law of the absolute maximum, similar to that found in the
context of Branching Brownian Motion. We give evidence that the random measure
$Z$ is a version of the derivative martingale associated with the continuum
Gaussian Free Field.
