  In this paper, we give a partial affirmative answer to the BAB conjecture for
$3$-folds in characteristic $p>5$. Specifically, we prove that a set
$\mathcal{D}$ of weak Fano $3$-folds over an uncountable algebraically closed
field is bounded, if each element $X \in \mathcal{D}$ satisfies certain
conditions regarding the Gorenstein index, a complement and Kodaira type
vanishing. In the course of the proof, we also study a uniform lower bound for
Seshadri constants of nef and big invertible sheaves on projective $3$-folds.
