We present a novel geometric construction of convergent and overconvergent p-adic Drinfeld modular forms of level 1 and Zp-weights. We reinterpret Hattori’s construction of convergent p-adic Drinfeld modular forms using our framework. Following the work of López and Petrov, we define convergent and overconvergent Drinfeld modular forms with A-expansions, and prove analogues of some standard theorems in this setting.