  Let $G \otimes _f H$ denote the Sierpi\'nski product of $G$ and $H$ with
respect to the function $f$. The Sierpi\'nski general position number ${\rm
gp}{_{\rm S}}(G,H)$ is introduced as the cardinality of a largest general
position set in $G \otimes _f H$ over all possible functions $f$. Similarly,
the lower Sierpi\'nski general position number $\underline{{\rm gp}}{_{\rm
S}}(G,H)$ is the corresponding smallest cardinality. The concept of
vertex-colinear sets is introduced. Bounds for the general position number in
terms of extremal vertex-colinear sets, and bounds for the (lower) Sierpi\'nski
general position number are proved. The extremal graphs are investigated.
Formulas for the (lower) Sierpi\'nski general position number of the \SP{s}
with $K_2$ as the first factor are deduced. It is proved that if $m,n\geq 2$,
then ${\rm gp}{_{\rm S}}(K_m,K_n) = m(n-1)$ and that if $n\ge 2m-2$, then
$\underline{{\rm gp}}{_{\rm S}}(K_m,K_n) = m(n-m+1)$.
