  Let $\mathcal{O}_K$ be a complete discrete valuation ring of mixed
characteristic $(0,p)$ with perfect residue field. We prove the existence of
the Hodge-Newton filtration for $p$-divisible groups over $\mathcal{O}_K$ with
additional endomorphism structure for the ring of integers of a finite,
possibly ramified field extension of $\mathbb{Q}_p$. The argument is based on
the Harder-Narasimhan theory for finite flat group schemes over
$\mathcal{O}_K$. In particular, we describe a sufficient condition for the
existence of a filtration of $p$-divisible groups over $\mathcal{O}_K$
associated to a break point of the Harder-Narasimhan polygon.
