  We generalize results about local heights previously proved in the case of
discrete absolute values to arbitrary non-archimedean absolute values of rank
1. First, this is done for the induction formula of Chambert-Loir and
Thuillier. Then we prove the formula of Burgos--Philippon--Sombra for the toric
local height of a proper normal toric variety in this more general setting. We
apply the corresponding formula for Moriwaki's global heights over a finitely
generated field to a fibration which is generically toric. We illustrate the
last result in a natural example where non-discrete non-archimedean absolute
values really matter.
