  We show a mathematically precise version of the SYZ conjecture, proposed in
the family Floer context, for the conifold with a conjectural mirror relation
between smoothing and crepant resolution. The singular T-duality fibers are
explicitly written and exactly correspond to the codimension-2 `missing points'
in the mirror cluster variety, which confirms the speculation of Chan,
Pomerleano, and Ueda but only in the non-archimedean setting. Concerning purely
the area of Berkovich geometry and forgetting all the mirror symmetry
background, our B-side analytic fibration is also a new explicit example of
affinoid torus fibration with singular extension.
