  A conjecture, known as the Shokurov-Koll\'ar connectedness principle,
predicts the following. Let $(X,B)$ be a pair, and let $f \colon X \rightarrow
S$ be a contraction with $-(K_X + B)$ nef over $S$; then, for any point $s \in
S$, the intersection $f^{-1} (s) \cap \mathrm{Nklt}(X,B)$ has at most two
connected components, where $\mathrm{Nklt}(X,B)$ denotes the non-klt locus of
$(X,B)$. This conjecture has been extensively studied in characteristic zero,
and it has been recently settled in that context. In this work, we consider
this conjecture in the setup of positive characteristic algebraic geometry. We
prove this conjecture holds for threefolds in characteristic $p> 5$, and, under
the same assumptions, we characterize the cases in which $\mathrm{Nklt}(X,B)$
fails to be connected.
