  In this paper we prove complete $p$-adic analogues of Kleinbock's theorems
\cite{Kleinbock-extremal, Kleinbock-exponent} on inheritance of Diophantine
exponents for affine subspaces. In particular, we answer in the affirmative
(and in a stronger form), a conjecture of Kleinbock and Tomanov \cite{KT}, as
well as a question of Kleinbock \cite{Kleinbock-exponent}. Our main innovation
is the introduction of a new $p$-adic Diophantine exponent which is better
suited to homogeneous dynamics, and which we show to be closely related to the
exponent considered by Kleinbock and Tomanov.
