  The motivic Mahowald invariant was introduced in \cite{Qui19a} and
\cite{Qui19b} to study periodicity in the $\mathbb{C}$- and
$\mathbb{R}$-motivic stable stems. In this paper, we define the motivic
Mahowald invariant over any field $F$ of characteristic not two and use it to
study periodicity in the $F$-motivic stable stems. In particular, we construct
lifts of some of Adams' classical $v_1$-periodic families \cite{Ada66} and
identify them as the motivic Mahowald invariants of powers of $2+\rho \eta$.
