A $p$-adic Second Main Theorem

Let $\mathbf{K}$ be an algebraically closed field of arbitrary characteristic, complete with respect to a non-archimedean absolute value $\mathrm{\mid }\mathrm{\mid }$. We establish a Second Main Theorem type estimate for analytic map $f\text{\hspace{0.17em}⁣}:\mathbf{K}\to {\mathbb{P}}^n(\mathbf{K})$ and a family of $n$ hypersurfaces in ${\mathbb{P}}^n(\mathbf{K})$ intersecting transversally and not all being hyperplanes. This implements the previous work of Levin where the case of all hypersurfaces having degree greater than one was studied.