  As a natural sequel for the study of $A$-motivic cohomology, initiated in
[Gaz], we develop a notion of regulator for rigid analytically trivial Anderson
$A$-motives. In accordance with the conjectural number field picture, we define
it as the morphism at the level of extension modules induced by the exactness
of the Hodge-Pink realization functor. The purpose of this text is twofold: we
first prove a finiteness result for $A$-motivic cohomology and, under a weight
assumption, we then show that the source and the target of regulators have the
same dimension. It came as a surprise to the author that the image of this
regulator might not have full rank, preventing the analogue of a renowned
conjecture of Beilinson to hold in our setting.
