Characters in p-adic Heisenberg and Lattice Vertex Operator Algebras

We study characters of states in $p$-adic vertex operator algebras. In particular, we show that the image of the character map for both the $p$-adic Heisenberg and $p$-adic lattice vertex operator algebras contain infinitely-many non-classical $p$-adic modular forms which are not contained in the image of the algebraic character map.