  Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic
irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we
define an analogue of the Hodge-Tate map which is suitable for the study of
Drinfeld modules over $\mathbf{F}_q[t]$ and, using it, develop a geometric
theory of $\wp$-adic Drinfeld modular forms similar to Katz's theory in the
case of elliptic modular forms. In particular, we show that for Drinfeld
modular forms with congruent Fourier coefficients at $\infty$ modulo $\wp^n$,
their weights are also congruent modulo $(q^d-1)p^{\lceil \log_p(n)\rceil}$,
and that Drinfeld modular forms of level $\Gamma_1(\mathfrak{n})\cap
\Gamma_0(\wp)$, weight $k$ and type $m$ are $\wp$-adic Drinfeld modular forms
for any tame level $\mathfrak{n}$ with a prime factor of degree prime to $q-1$.
