  Our objective in this series of two articles, of which the present article is
the first, is to give a Perrin-Riou-style construction of $p$-adic
$L$-functions (of Bella\"iche and Stevens) over the eigencurve. As the first
ingredient, we interpolate the Beilinson-Kato elements over the eigencurve
(including the neighborhoods of $\theta$-critical points). Along the way, we
prove \'etale variants of Bella\"iche's results describing the local properties
of the eigencurve. We also develop the local framework to construct and
establish the interpolative properties of these $p$-adic $L$-functions away
from $\theta$-critical points.
