Let $p$ be a fixed prime.

Let $A$ be an abelian variety of $\mathbb{Q}_p$ and let $\mathcal{A}$ denote the Neron model of $A$ over $\mathbb{Z}_p$. Now, the multiplication by $p$ map in $\text{End}(A)$ extends to an endomorphism of $\mathcal{A}$, and let $\mathcal{A}[p]$ denote the (scheme-theoretic) kernel of this morphism.

Here are my questions on the properties of $\mathcal{A}[p]$:

- Is it always quasi-finite?
- Is it always flat?
- Is it always separated?

If $\mathcal{A}$ is a semi-abelian scheme, then $\mathcal{A}[p]$ is a quasi-finite flat separated group scheme by SGA 7, Exp IX, Lemme 2.2.1. I want to know what happens when $\mathcal{A}$ is not semi-abelian. For instance, what can we say about $\mathcal{E}[p]$ if $E$ is an elliptic curve over $\mathbb{Q}_p$ which has additive reduction?