  We extend Gubler--K\"unnemann's theory of $\delta$-forms from algebraic
varieties to good Berkovich spaces. This is based on the observation that
skeletons in such spaces satisfy a tropical balance condition. Our main result
is that complete intersection formal models of cycles give rise to Green
$\delta$-forms for their generic fibers. We moreover show that, in certain
situations, intersection numbers on formal models are given by the
$\star$-product of their Green currents. In this way, we generalize some
results for divisor intersection to higher codimension situations. We
illustrate the mentioned results in the context of an intersection problem for
Lubin--Tate spaces.
