Counterexample. Let $$P=(\mathbb Q\times\{0,1\})\cup\{-\infty,+\infty\}$$ be ordered so that $$-\infty\lt(x,i)\lt+\infty$$ for $(x,i)\in\mathbb Q\times\{0,1\}$, and $$(x,i)\lt(x',i')\iff x\lt x'$$ for $(x,i),(x',i')\in\mathbb Q\times\{0,1\}$.

Counterexample. Let $$P=\{(x,i)\in\mathbb Q\times\{0,1\}:0\le x\le1,\ x\ne i\}$$ be ordered so that $$(x,i)\lt(x',i')\iff x\lt x'.$$