I recently looked at some of the work of Nicaise on non-archimedean SYZ, and at the end of this paper arxiv.org/pdf/1708.09637 he constructs $E^{an}$ for $E$ a Tate curve. There is a retraction $\rho : E^{an} \rightarrow S^1 $ given by collapsing the trees down to the circle. The author then remarks that this is the "simplest case of a non-archimedean SYZ fibration".
In what sense is this map a fibration? The fibres are either singleton points, or a tree, unless I have misunderstood this retraction map. In the usual case of SYZ fibrations for a torus, we'd expect the fibres to be copies of $S^1$.
I am new to studying this area so apologies if I am missing something obvious.
