Let $A\subseteq K$ be discrete valuation ring in its field of fractions. Let $\kappa:=A/\mathfrak m$ be the residue field. Let $X$ be a scheme, flat and projective over $A$. Consider a short exact sequence of coherent $\mathcal O_X$-modules, which are flat over $A$ $$0\to\mathcal F_1\to \mathcal F\to\mathcal F_2\to0$$
If we have that $$0\to\mathcal F_1\otimes_AK\to\mathcal F\otimes_AK\to\mathcal F_2\otimes_AK\to0$$ does not split, then can we also have that $$0\to\mathcal F_1\otimes_A\kappa\to\mathcal F\otimes_A\kappa\to\mathcal F_2\otimes_A\kappa\to0$$ does not split?