Formal completion of modular curves

Question

Let $N\geq 4$ be an integer coprime with $p$, where $p$ is a fixed prime number. Then we know that there exists a scheme $Y_N$ over $\text{Spec}(\mathbb{Z}_p)$ whose $\text{Spec}(R)$-points are elliptic curves over $R$ with a level $N$-structure, where $R$ is a $\mathbb{Z}_p$-algebra. $Y_N$ is the non-compactified modular curve of level $\Gamma_1(N)$. We also know, by representability, that there exists a universal elliptic curve $\mathcal{E}$ over $Y_N$. My question is the following. What does it happen if we now $p$-adically complete the picture, i.e. if we consider the formal scheme $\mathcal{Y}_N$ over $\text{Spf}(\mathbb{Z}_p)$ given by completion, does it still have a moduli interpretation? Of course we can complete also $\mathcal{E}$, but now it's no more an elliptic curve, it's a kind of formal elliptic curve. Is it still true that a morphism $\text{Spf}(R)\rightarrow\mathcal{Y}_N$ gives an elliptic curve (formal or real?) over $R$ with a $N$-level structure?