  We study the extremal properties of the "integer-valued Gaussian" a.k.a.\
DG-model on the hierarchical lattice $\Lambda_n:=\{1,\dots,b\}^n$ (with
$b\ge2$) of depth $n$. This is a random field $\varphi\in\mathbb Z^{\Lambda_n}$
with law proportional to
$e^{\frac12\beta(\varphi,\Delta_n\varphi)}\prod_{x\in\Lambda_n}\#(d\varphi_x)$,
where $\Delta_n$ is the hierarchical Laplacian, $\beta$ is the inverse
temperature and $\#$ is the counting measure on $\mathbb Z$. Denoting
$\beta_c:=2\pi^2/\log b$ and $m_n:=\beta^{-1/2}[(2\log b)^{1/2}n-\frac32(2\log
b)^{-1/2}\log n]$, for $0<\beta<\beta_c$ we prove that, along increasing
sequences of $n$ such that the fractional part of $m_{n}$ converges to an
$s\in[0,1)$, the centered maximum $\max_{x\in\Lambda_n}\varphi_x-\lfloor
m_n\rfloor$ tends (as $n\to\infty$) in law to a discrete variant of a randomly
shifted Gumbel law with the shift depending non-trivially on $s$. The
convergence extends to the extremal process whose law tends to a decorated
Poisson point process with a random intensity measure. The proofs rely on
renormalization-group analysis which enables a tight coupling of the DG-model
to a Gaussian Free Field. The interval $(0,\beta_c]$ marks the full range of
values of $\beta$ for which the renormalization-group iterations tend to a
"trivial" fixed point.
