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We construct an Asplund Banach space $\mathcal{X}$ with a normingMarku\v{s}evi\v{c} basis such that $\mathcal{X}$ is not weakly compactlygenerated. This solves a long-standing open problem from the early nineties,originally due to Gilles Godefroy. En route to the proof, we construct apeculiar example of scattered compact space, that also solves a question due toWies{\l}aw Kubi\'s and Arkady Leiderman.

An Asplund space with norming Marku\v{s}evi\v{c} basis that is not weakly compactly generated