Schwartz spaces on L-monoids: non-Archimedean

We complete the Braverman-Kazhdan-Ng^{o} program over non-Archimedean local fields assuming local Langlands conjecture for tempered representations and a natural assumption on the $\gamma$-factors of non-supercuspidal discrete series. In particular, the program is complete unconditionally for general linear groups over non-Archimedean local fields.