Let $K$ be a non Achimedian valued field with valuation ring $\mathcal{O}$. Let futhermore $X$ be a projective scheme over $\mathcal{O}$ with map $X \to Spec(\mathcal{O})$.
Donote by $\widetilde{X}$ the pullback of $X$ along $Spec(K) \to Spec(\mathcal{O})$.
My question is if (and why) we can identify the $K$-valued points $\widetilde{X}(K)= Hom(Spec(K),\widetilde{X})$ with "$\mathcal{O}$-valued points" $X(\mathcal{O})$? Is there a lifting theorem in the game?