I personally would say that the root of the problem is the absence of a globally defined exponential map on $\mathbb{C}_p$. In the complex world $z \mapsto q(z) = \exp(2 \pi i z)$ is an isomorphism between $\mathbb{C} / \mathbb{Z}$ and $\mathbb{C}^\times$; but $\mathbb{C}_p / \mathbb{Z}$ is a horrible mess, and has nothing to do with $\mathbb{C}_p^\times$.

This makes it hard to bridge between the world of $q$-expansions and the world of analytic functions satisfying nice functional equations. The functional equation of a modular form looks nice if you write it in terms of $z$, but if you try and write it in terms of $q$ alone, it will make no sense. 

That's why one has to take a different approach, building $p$-adic modular forms directly as $p$-adic analytic functions (more precisely: analytic sections of line bundles) on modular curves (more precisely: the rigid spaces attached to modular curves), rather than trying to build them as functions on some other space satisfying functional equations which have the *post facto* consequence that they descend to modular curves, as in the theory over $\mathbb{C}$.

(You might be interested to know that the power series in $q$ defining certain natural modular forms do converge $p$-adically for $q$ < 1, and this is very important in Tate's theory of uniformization of elliptic curves over $\mathbb{Q}_p$; but these series aren't $p$-adic modular forms as such.)