A compact hyperbolic surface of genus
g
g
is said to be extremal if it admits an extremal disc, a disc of the largest radius determined by
g
g
. We know how many extremal discs are embedded in a non-orientable extremal surface of genus
g
=
3
g=3
or
g
>
6
g>6
. We show in the present paper that there exist
144
144
non-orientable extremal surfaces of genus
4
4
, and find the locations of all extremal discs in those surfaces. As a result, each surface contains at most two extremal discs. Our methods used here are similar to those in the case of
g
=
3
g=3
.