It is well known that a modular form (of weight $k$ and level $N$) is in particular also a classical modular form; this can be seen both using Serre's definition with $q$-expansion and the Katz's one, viewing a modular form as function of marked elliptic curves.
My question is if there is some kind of converse of this fact. In particular starting with a $p$-adic modular form $f$ and viewing it as function on ordinary (at $p$) marked elliptic curves defined over $\mathbb{C}_p$ or, equivalently as rigid analytic section of the sheaf $\underline{\omega}^k$ over the ordinary locus of the modular curve $X_0(N)(\mathbb{C}_p)$, one can try to restrict it to a function on some open subset of the classical upper half plane. In other words, I don't require it to be a global rigid analytic section of $X_0(N)(\mathbb{C})$, i.e. a classical modular form, but only a holomorphic function on some open. Is, maybe, this possible in the case where $f$ is a rigid analytic section not only over the ordinary locus, but on some wide open neighbourhood?
In particular I am interested in the case where $f$ is the Colemann primitive of the $p$-depletion of a cusp form; in this case it is proved that indeed $f$ is a rigid analytic section over some wide open neighbourhood.