Historically, the result is due to Hervé Jacquet

MR0369624 (51 #5856)
Jacquet, Hervé Sur les représentations des groupes réductifs p-adiques.
C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Aii, A1271–A1272.
22E50

(From MathSciNet)

"Let F denote a nonarchimedean nondiscrete locally compact field and G a reductive F-group. A representation r or G(F) in a complex vector space V is said to be "smooth'' if the stabilizer of each vector in V is open in G(F); r is said to be "admissible'' if it is smooth and the space of vectors in V fixed by any open compact subgroup of G(F) is finite-dimensional. The author proves that every irreducible smooth representation of G(F) is automatically admissible. The proof, like the result itself, is surprisingly simple and clever. "