Let $A$ be a smooth commutative $k$-algebra, for $k$ a commutative ring. By the Hochschild-Kostant-Rosenberg theorem, we have that $HH^*_k(A)\cong \Lambda^* \mathrm{Der}_k(A,A)$, where $\mathrm{Der}_k(A,A)$ is the $A$-module of $k$-linear $A$ derivations. Recall that $HH^*_k(A)$ is an graded-commutative algebra under cup product (it is actually a Gerstenhaber algebra, but we only need the cup product for now).
If $k$ is a field with characteristic not 2, it is easy to see that the relation $[f]\smallsmile[f]=0$ for $[f]\in HH^1_k(A)$ is satisfied, as $[f]\smallsmile[f]$ is 2-torsion. On the other hand, some more work seems to be required in the characteristic 2: even given $A=\mathbb{F}_2[x,y]$, it is not immediately obvious to me (and perhaps I just haven't played around with this enough) how to decompose $[\frac{\partial}{\partial x}]\smallsmile[\frac{\partial}{\partial x}]$ as a sum of Hochschild 2-boundaries, ie functions of the form $f(a,b)=a\cdot g(b)- g(ab)+g(a)\cdot b$. Does anyone know such a decomposition?