(1.4) The aim of the present paper is to construct the non-archimedean SYZ fibration in full generality, and to prove some of its conjectural properties. This paves the way for a better understanding of the Gromov-Hausdorff limits and the SYZ conjecture. Our construction of the SYZ fibration builds upon the original work of Kontsevich and Soibelman and the relations with the Minimal Model Program discovered by the first two authors in [NX16a]. This discovery has led to a surprising dictionary where the SYZ heuristic can be translated into precise predictions about the structure of minimal models, which can then be proven with techniques from the Minimal Model Program – see for instance [KX16] and [NX16b]. Our main new result here is that the non-archimedean SYZ fibration is a smooth affinoid torus fibration away from a codimension two subset of the base (Theorem 6.1), as implied by Conjectures 1 and 3 in [KS06]. This amounts to proving that minimal dlt models with reduced special fiber of Calabi-Yau varieties are snc along the one-dimensional strata of the special fiber (Theorem 4.5), and have a toric structure along these strata (Proposition 5.4).