An example concerning set addition in $\mathbb F_2^n$

We construct sets $A$ and $B$ in a vector space over $\mathbb F_2$ with the property that $A$ is “statistically” almost closed under addition by $B$ in the sense that $a + b$ almost always lies in $A$ when $a\in A$ and $b\in B$, but which is extremely far from being “combinatorially” almost closed under addition by $B$: if $A'\subset A$, $B'\subset B$ and $A' + B'$ is comparable in size to $A'$, then $|B'|\lessapprox |B|^{1/2}$.