We overview some of the foundations of the so-called henselian rigid geometry, and show that henselian rigid geometry has many aspects, useful in applications, that one cannot expect in the usual rigid geometry. This is done by announcing a few characteristic results, one of which is an analogue of Zariski Main Theorem.
"On Henselian Rigid Geometry." Taiwanese J. Math. 21 (3) 531 - 547, 2017. https://doi.org/10.11650/tjm/7989