We show that all extremal elliptic surfaces in characteristic 2 and 3 are
obtained from rational extremal elliptic surfaces as purely inseparable base extensions.
As a corollary, we can show that the automorphism group of every supersingular elliptic
$K3$ surface has an element of infinite order which acts trivially on the global sections of
the sheaf of differential forms of degree 2. We also determine the structures of Mordell-
Weil groups for extremal rational elliptic surfaces in these characteristics.