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We construct a normal projective rigid analytic compactification of an arbitrary Drinfeld modular variety whose boundary is stratified by modular varieties of smaller dimensions. This generalizes work of Kapranov. Using an algebraic modular compactification that generalizes Pink and Schieder's, we show that the analytic compactification is naturally isomorphic to the analytification of Pink's normal algebraic compactification. We interpret analytic Drinfeld modular forms as the global sections of natural ample invertible sheaves on the analytic compactification and deduce finiteness results for spaces of such modular forms.
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