  This is a paper in a series to study vertex algebra-like structures arising
from various algebras including quantum affine algebras and Yangians. In this
paper, we study notions of $\hbar$-adic nonlocal vertex algebra and
$\hbar$-adic (weak) quantum vertex algebra, slightly generalizing
Etingof-Kazhdan's notion of quantum vertex operator algebra. For any
topologically free $\C[[\h]]$-module $W$, we study $\hbar$-adically compatible
subsets and $\hbar$-adically $\S$-local subsets of $(\End W)[[x,x^{-1}]]$. We
prove that any $\hbar$-adically compatible subset generates an $\hbar$-adic
nonlocal vertex algebra with $W$ as a module and that any $\hbar$-adically
$\S$-local subset generates an $\hbar$-adic weak quantum vertex algebra with
$W$ as a module. A general construction theorem of $\hbar$-adic nonlocal vertex
algebras and $\hbar$-adic quantum vertex algebras is obtained. As an
application we associate the centrally extended double Yangian of $\sl_{2}$ to
$\hbar$-adic quantum vertex algebras.
