A question regarding unique matrix factorization

Question

Let $B$ be an $n\times\ell$ binary matrix with $n>\ell$, and rank $\ell$. Define $A=BB^T$. Given the rank $\ell$ matrix $A\in\mathbb{N}^{n\times n}$, and assuming that $A$ is such that there exists a binary matrix $B$ such that $A=BB^T$, is it always possible to uniquely find $B$, up to permutations of columns/rows? It seems as a rank factorization problem, but I am not sure how to prove uniqueness under this setting.

Answer 1

Take $3I_2$. You will never be able to to find any binary matrix to satisfy your equation.