  In earlier papers (A. N. Kochubei, Pacif. J. Math., 269 (2014), 355-369; J.
Math. Anal. Appl.483 (2020), Article 123609), one of the authors developed a
theory of pseudo-differential equations for radial real-valued functions on a
non-Archimedean local field, with some features resembling those of classical
ordinary differential equations. Here we consider equations of this kind, but
with a weak degeneration. Under various assumptions, we prove the local
existence and uniqueness of mild solutions, existence of their global
extensions and a regularity property.
