You must have $p=2$ otherwise antisymmetry implies the alternate identity $[x,x]=0$.
The minimal example taylored on the pattern below I can provide is the (multiplicative) subalgebra (over the field $\mathbb{Z}/2\mathbb{Z}$) generated by
$$
E_{12}+E_{23}=
\begin{pmatrix}
0 & 1 & 0\\
0 & 0 & 1\\
0 & 0 & 0
\end{pmatrix}\ ;\
E_{13}=
\begin{pmatrix}
0 & 0 & 1\\
0 & 0 & 0\\
0 & 0 & 0
\end{pmatrix}
$$
then, you can check you have your algebra.
Now, if, for some reason, you need to preserve some identities or relations of an existing Lie algebra (always in characteristic $2$ for the reason said before), proceed as follows
- take you Lie algebra $(L,[-,-]_L)$ over $k$ (of characteristic $2$)
- add two dimensions by forming $L^{(1)}=L\oplus k.e_1\oplus k.e_2$
- extend the bracket $[x,y]$ to $L^{(1)}$ by
- $[x,y]_L$ if $x,y\in L$
- $[x,e_i]=[e_i,x]=0$ if $x\in L$
- $[e_1,e_1]=e_2;\ [e_2,e_i]=[e_i,e_2]=0$
then $L^{(1)}$ admits $L$ as a sector (direct summand and quotient), fulfills anti-symmetry and Jacobi, but not the alternate identity ($[x,x]=0$).