Bourguignon showed that if a compact manifold does not admit positive scalar curvature metrics, then any scalar flat metric (actually, any non-negative scalar curvature metric) is Ricci-flat; I suppose this is what you mean when you write "strongly scalar flat". But in three dimensions, the Ricci curvature determines the full curvature tensor, in particular, a Ricci-flat metric is flat. So any non-flat three-manifold which does not admit positive scalar curvature metrics will provide an example. 

If I'm not mistaken, Gromov and Lawson proved that a compact three-manifold admits positive scalar curvature if and only if its prime decomposition contains no aspherical factors; note, this was before the Poincaré conjecture had been verified, so there would have been a caveat at the time of publication.

So $T^3\#S^1\times S^2$ is an example of a compact three-manifold of type (C). Note, $T^3\#S^1\times S^2$ is not flat as a non-trivial connected sum of compact manifolds of dimension at least three is never aspherical, but flat $n$-dimensional manifolds have universal cover $\mathbb{R}^n$.

In dimension four, you can sometimes use Seiberg-Witten invariants to rule out the existence of positive scalar curvature metrics, and then use the Hitchin-Thorpe inequality to rule out the existence of a Ricci-flat metric. For example,   a compact Kähler surface with $b^+ \geq 2$ does not admit positive scalar curvature metrics; blowing up doesn't change this, but it eventually violates the Hitchin-Thorpe inequality.

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A proof of Bourguignon's result can be found in Kazdan and Warner's paper *Prescribing Curvatures*, namely Lemma 5.2. As for the result of Gromov and Lawson, see Chapter IV, Theorem 6.18 of Lawson and Michelsohn's *Spin Geometry* and the discussion which follows.