= Solution
Let $H\leq\mathbb F_p^n$ be a subspace with $|H|$ much larger than $K^2$, and choose linearly independent vectors $e_1,\ldots,e_{K-1}$ whose images are independent modulo $H$. Set
$$
A=H\cup\{e_1,\ldots,e_{K-1}\}.
$$
Then $|A|=|H|+K-1\sim|H|$, while $A+A$ is the union of $H$, the $K-1$ disjoint cosets $e_i+H$, and at most $K^2$ exceptional sums $e_i+e_j$. Hence
$$
|A+A|\sim K|A|.
$$
Every subspace containing $A$ must contain $H$ and all $e_i$, so it has at least $p^{K-1}|H|\sim p^{K-1}|A|$ elements. Thus the exponential dependence on $K$ in the <Freiman-Ruzsa theorem over a finite field> cannot in general be replaced by a subexponential one.
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