A double extension ($mathscr{D}$ extension) of a Lie (super)algebra $mathfrak a$ with a non-degenerate invariant symmetric bilinear form $mathscr{B}$, briefly: a NIS-(super)algebra, is an enlargement of $mathfrak a$ by means of a central extension and a derivation; the affine Kac-Moody algebras are the best known examples of double extensions of loops algebras. Let $mathfrak a$ be a restricted Lie (super)algebra with a NIS $mathscr{B}$. Suppose $mathfrak a$ has a restricted derivation $mathscr{D}$ such that $mathscr{B}$ is $mathscr{D}$-invariant. We show that the double extension of $mathfrak a$ constructed by means of $mathscr{B}$ and $mathscr{D}$ is restricted. We show that, the other way round, any restricted NIS-(super)algebra with non-trivial center can be obtained as a $mathscr{D}$-extension of another restricted NIS-(super)algebra subject to an extra condition on the central element. We give new examples of $mathscr{D}$-extensions of restricted Lie (super)algebras, and pre-Lie superalgebras indigenous to characteristic 3.
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