= Solution
Give $S^2\times\mathbb R$ the <Riemannian product> of the unit round metric and the Euclidean metric. It is complete and has infinite <diameter of a metric space>[diameter]. The round sphere has <scalar curvature> $2$, while the line has scalar curvature $0$; scalar curvature is additive under Riemannian products, so
$$
\operatorname{Scal}_{S^2\times\mathbb R}=2.
$$
Thus this manifold has a strictly positive uniform lower bound on scalar curvature but violates the conclusion of the <Bonnet-Myers theorem>. Its <Ricci curvature> vanishes in the $\mathbb R$ direction, showing precisely why a scalar-curvature bound is insufficient.
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