The most elementary way to define $p$-adic modular forms is via limits of classical modular forms.
More precisely $f \in \mathbb{Z}\_p[[q]]$ is called a $p$-adic modular form
if there are modular forms $f\_n$ with integral coefficients such that
$f \equiv f\_n \mod p^n$ (as $q$-expansions). Note it does not really make sense to attribute 'a weight' to $f$ since 
the $f\_n$ are allowed to have different (increasing weights). This is the older definition by Serre.


I know (but I do not understand) a newer definition by Katz, which has a more geometric flavor.
See here math.arizona.edu/~swc/notes/files/01BuzzardL2.pdf.


So we have an approach using the $q$-expansion and we have an approach to $p$-adic modular forms
using geometric ideas.


My question now is, whether there is also a developed theory on analytic p-adic modular forms?


Some ideas what this might mean. For example we could consider the Eisenstein series
$$ E\_4(\tau)=\sum\_{n,m \in \mathbb{Z}} \frac{1}{(n\tau+m)^4} $$
as function of $\tau$ not being an element of the upper half-plane but of some subset of $\mathbb{C}\_p$.


Does this sum even converge somewhere in $\mathbb{C}\_p$. And is it (up to a constant) a classical or Katz $p$-adic modular
form? Does it even equal (mod p) the Eisenstein series $E\_4$?


A similar question could be posed for the infinite product usually defining the $\Delta$-function
$$ q\prod\_{n=1}^\infty (1-q^n)^{24}. $$
Does this converge somewhere when $q$ is in some subset of $\mathbb{C}\_p$. Is it a $p$-adic modular form?


If there is no such theory? Why not? Is it not interesting?



Are there 'analytic' $p$-adic modular forms.
There is such a theory, but the analytic object that the forms live on is an analytified modular curve, not simply $\mathbb{C}\_p$ (though there is a "$p$-adic upper-half plane" that can be used to uniformize some similar moduli spaces, but as far as I know not the usual modular curves).


Basically, if $f$ is a classical modular form of some weight $k$, $f$ can be realized as a section of a sheaf $\omega^{\otimes k}$ on a complex-analytic modular curve such as $X\_1(N)$ obtained via quotient from the complex upper-half plane. These curves and sheaves have algebraic models defined over $\mathbb{Q}$, and (under mild hypotheses) the form $f$ actually arises from a section of the associated sheaf $\omega^{\otimes k}$ on the modular curve defined over $\mathbb{Q}$ (or some finite extension).


Now you can go in another direction and consider the rigid-analytic space over $\mathbb{Q}\_p$ associated to the smooth algebraic curve $X\_1(N)$ and your form $f$ gives rise to a $p$-adic analytic object on this curve. Now one can play games like considering subspaces obtained by removing disks around supersingular points to obtain general $p$-adic modular forms (such as limits of classical forms of varying weight) and overconvergent modular forms.


For what it's worth, these curves are $p$-adic analytic moduli spaces, and this point of view on $p$-adic modular forms essentially differs from Katz's by thinking about rigid spaces as opposed to formal schemes (the Raynaud point of view on $p$-adic analytic geometry).


Look at the papers of Coleman (such as his $p$-adic Banach spaces paper or the eigencurve paper with Mazur) if you would like to read more on this point of view.


