  We define normalized versions of Berkovich spaces over a trivially valued
field $k$, obtained as quotients by the action of $\mathbb R_{>0}$ defined by
rescaling semivaluations. We associate such a normalized space to any special
formal $k$-scheme and prove an analogue of Raynaud's theorem, characterizing
categorically the spaces obtained in this way. This construction yields a
locally ringed $G$-topological space, which we prove to be $G$-locally
isomorphic to a Berkovich space over the field $k((t))$ with a $t$-adic
valuation. These spaces can be interpreted as non-archimedean models for the
links of the singularities of $k$-varieties, and allow to study the birational
geometry of $k$-varieties using techniques of non-archimedean geometry
available only when working over a field with non-trivial valuation. In
particular, we prove that the structure of the normalized non-archimedean links
of surface singularities over an algebraically closed field $k$ is analogous to
the structure of non-archimedean analytic curves over $k((t))$, and deduce
characterizations of the essential and of the log essential valuations, i.e.
those valuations whose center on every resolution (respectively log resolution)
of the given surface is a divisor.
