Let $k$ be a field of characteristic $p$, and let $L$ be a restricted Lie algebra over $k$. Thus $L$ is a lie algebra together with a map $(-)^{[p]}:L\to L$ satisfying the three axioms found here.
Call $L$ nil if there is some $n$ such that $x^{[p]^n}=0$ for all $x\in L$. Here $x^{[p]^n}$ signifies the image of $x$ under the map $(-)^{[p]}$ composed with itself $n$ times.
Suppose now that $L$ is finite-dimensional. Engel's theorem and the first axiom listed above show that if $L$ is nil, then $L$ is nilpotent. Is the converse true? Here is my idea so far.
If $v_1,\ldots,v_m$ form a basis for $L$, let $n_i$ be such that $v_i^{[p]^{n_i}}=0$ for $i=1,\ldots,m$. It seems that we should be able to use the $n_i$ to construct a large enough number $n$ so that $x^{[p]^n}=0$ for any $x\in L$. The trouble comes from using the axioms to expand terms of the form $(v_i+v_j)^{[p]^k}$.
In case the converse is not true, it might make sense to search for a counterexample. One such would be to find a restricted Lie algebra $L\subset\mathfrak{gl}_m(k)$ such that for each $x\in L$, there is some $n=n(x)$ such that $x^{p^{n(x)}}=0$, with
$$\sup_{x\in L}\{n(x)\}=\infty$$
Notice here that the map $(-)^{[p]}$ is just the $p$th matrix power map.