  In this paper we prove that every finite-dimensional nilpotent restricted Lie
algebra over a field of prime characteristic has an outer restricted derivation
whose square is zero unless the restricted Lie algebra is a torus or it is
one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra
in characteristic two as an ordinary Lie algebra. This result is the restricted
analogue of a result of T\^og\^o on the existence of nilpotent outer
derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and
the Lie-theoretic analogue of a classical group-theoretic result of Gasch\"utz
on the existence of $p$-power automorphisms of $p$-groups. As a consequence we
obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra
has an outer restricted derivation.
