  We establish existence theorems for the image of the normalized character map
of the $p$-adic Heisenberg algebra $S$ taking values in the algebra of Serre
$p$-adic modular forms $M_p$. In particular, we describe the construction of an
analytic family of states in $S$ whose character values are the well-known
$\Lambda$-adic family of $p$-adic Eisenstein series of level one built from
classical Eisenstein series. This extends previous work treating a
specialization at weight $2$, and illustrates that the image of the character
map contains nonzero $p$-adic modular forms of every $p$-adic weight. In a
different direction, we prove that for $p=2$ the image of the rescaled
character map contains every overconvergent $2$-adic modular form of weight
zero and tame level one; in particular, it contains the polynomial algebra
$\mathbf{Q}_2[j^{-1}]$. For general primes $p$, we study the square-bracket
formalism for $S$ and develop the idea that although states in $S$ do not
generally have a conformal weight, they can acquire a $p$-adic weight in the
sense of Serre.
