The answer is Yes. Consider e.g. the restricted simple Lie algebra $W(1)$ over a field $\mathbb{F}$ of characteristic $p>3$ and let $\hat{W}(1)$ denote its universal central extension. As $H^2(W(1), \mathbb{F})\neq 0$, $\hat{W}(1)$ is a non-trivial central extension of $W(1)$. Moreover, as all derivations of $W(1)$ are inner, denoted by $Z(\hat{W}(1))$ the center of $\hat{W}(1)$, one has

$$D(\hat{W}(1))\cong D(W(1))\cong W(1)\cong \frac{\hat{W}(1)}{Z(\hat{W}(1))},$$

thus all derivations of $\hat{W}(1)$ are inner.

I realized that it is indeed possible. Consider e.g. the restricted simple Lie algebra $W(1)$ over a field $\mathbb{F}$ of characteristic $p>3$ and let $\hat{W}(1)$ denote its universal central extension. As $H^2(W(1), \mathbb{F})\neq 0$, $\hat{W}(1)$ is a non-trivial central extension of $W(1)$. Moreover, as all derivations of $W(1)$ are inner, denoted by $Z(\hat{W}(1))$ the center of $\hat{W}(1)$, one has

$$D(\hat{W}(1))\cong D(W(1))\cong W(1)\cong \frac{\hat{W}(1)}{Z(\hat{W}(1))},$$

thus all derivations of $\hat{W}(1)$ are inner.

The answer is Yes. Consider e.g. the restricted simple Lie algebra $W(1)$ over a field $\mathbb{F}$ of characteristic $p>3$ and let $\hat{W}(1)$ denote its universal central extension. As $H^2(W(1), \mathbb{F})\neq 0$, $\hat{W}(1)$ is a non-trivial central extension of $W(1)$. Moreover, as all derivations of $W(1)$ are inner, denoted by $Z(\hat{W}(1))$ the center of $\hat{W}(1)$, one has

$$D(\hat{W}(1))\cong D(W(1))\cong W(1)\cong \frac{\hat{W}(1)}{Z(\hat{W}(1))},$$

thus all derivation of $\hat{W}(1)$ are inner.

The answer is Yes. Consider e.g. the restricted simple Lie algebra $W(1)$ over a field $\mathbb{F}$ of characteristic $p>3$ and let $\hat{W}(1)$ denote its universal central extension. As $H^2(W(1), \mathbb{F})\neq 0$, $\hat{W}(1)$ is a non-trivial central extension of $W(1)$. Moreover, as all derivations of $W(1)$ are inner, denoted by $Z(\hat{W}(1))$ the center of $\hat{W}(1)$, one has

$$D(\hat{W}(1))\cong D(W(1))\cong W(1)\cong \frac{\hat{W}(1)}{Z(\hat{W}(1))},$$

thus all derivations of $\hat{W}(1)$ are inner.