  Real-valued differential forms on Berkovich analytic spaces were introduced
by Chambert-Loir and Ducros in 'Formes diff\'erentielles r\'eelles et courants
sur les espaces de Berkovich' using superforms on polyhedral complexes. We
prove a Poincar\'e lemma for these superforms and use it to also prove a
Poincar\'e lemma for real-valued differential forms on Berkovich spaces. For
superforms we further show finite dimensionality for the associated de Rham
cohomology on polyhedral complexes in all (bi-)degrees. We also show finite
dimensionality for the real-valued de Rham cohomology of the analytification of
an algebraic variety in some bidegrees.
