Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $\mathbb{F}$. Say I have a (projective) quadric $Q$ which is smooth over $\mathcal{O}_K$, such that the reduction $\bar{Q}$ (smooth over $\mathbb{F}$) contains a line $\cong \mathbb{P}^1$ defined over $\mathbb{F}$. Does it follow that $Q$ contains a line $\cong \mathbb{P}^1$ defined over $\mathcal{O}_K$?

If yes, I guess this would somehow follow from Hensel's lemma, but I'm not sure how exactly.