Let $L$ be a finite-dimensional Lie algebra over a field of characteristic zero. It is not difficult to see (and also follows from Theorem 4.4 of [G. Hochschild: Semi-simple algebras and generalized derivations, Amer. J. Math. 64, (1942), 677–694]) that the derivation algebra $D(L)$ of $L$ is simple if and only if so is $L$.

Now suppose that $L$ is defined over an algebraically closed field $\mathbb{F}$ of positive characteristic. If $D(L)$ is simple then all derivations of $L$ are inner (in particular, $L$ is restricted), and $L$ is either simple or a central extension of a simple Lie algebra. It is clear that such an extension cannot be split. Thus my question is the following:

Suppose that $L$ is a non-split finite-dimensional central extension of a restricted simple Lie algebra $\mathfrak{g}$ over a field $\mathbb{F}$ of characteristic $p>0$. Is it possible that every derivation of $L$ is inner?

Modular Lie Algebras(Springer, 1967) is now half a century old, you might look at his survey V.5 on derivations, along with the references. $\endgroup$ – Jim Humphreys Mar 8 '17 at 20:28