Let $p$ be a prime with $p\equiv 1\mod{3}$ (there are infinitely many of these). Then $-3$ is a quadratic residue mod $p$, so take a solution $w_1\in\mathbb{Z}$ of $X^2+X+1 \equiv 0\mod{p}$. Then $(2w_1+1)^2 \equiv -3\not\equiv 0 \mod{p}$, so $w_1$ is not a solution to $(X^2+X+1)'= 2X+1\equiv 0\mod{p}$. Hensel's lemma on $f(X) = X^2+X+1$ hence yields a $w_k\in\mathbb{Z}$ with $w_1 \equiv w_k\mod{p}$ and $w_k^2+w_k+1\equiv 0\mod{p^k}$. That $w_k \equiv w_1 \not\equiv \pm 1\mod{p}$, is easily checked, so we should only verify that the order of $w_k$ mod $p$ is the same as mod $p^k$. But we have $w_k\not\equiv 1\mod{p}$, $w_k^2 \not\equiv 1\mod{p}$, and $w_k^3\equiv 1\mod{p}$, so the order of $w_k$ mod $p$ is $3$, and the same argument shows this is true mod $p^k$.