Let $p$ be a prime. Let $K/\mathbb{Q}_p$ be a finite extension and $\mathcal{O}=\mathcal{O}_K$ its ring of integers.

Let $A_K$ be an abelian variety and $\mathcal{A}_\mathcal{O}$ denote its Neron model over $\mathcal{O}$. We don't assume that $\mathcal{A}_\mathcal{O}$ has good or semistable reduction.

Let $\ell$ be a prime different from $p$ and let $G=A[\ell]_\mathcal{O}$ be the scheme-theoretic kernel of multiplication by $\ell$ in $\mathcal{A}_\mathcal{O}$. We know that $G\times_{\text{Spec} ~\mathcal{O}} \text{Spec} ~K = A_K[\ell]$ is a finite subgroup scheme of order $\ell^{2g}$, where $g$ is the dimension of $A_K$.

I have a very stupid question here:

**Is $A[\ell]_{\mathcal{O}}$ finite and flat? (or at least quasi-finite)?
If not, are there some necessary conditions for $A[\ell]_{\mathcal{O}}$ to be finite and flat?**