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From Conrad's notes on rigid geometry:

More specifically, Gabber has given an example of a sheaf of modules $F$ on the closed unit disk $B^1$ such that $F$ is locally a direct limit of coherent sheaves but with nonzero degree-1 sheaf cohomology, so $F$ cannot be expressed as a direct limit of coherent sheaves over the entire affinoid space...

What is this example due to Gabber? Is there a reference describing it?

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  • $\begingroup$ FWIW, the user appears to have been part of a group of accounts controlled by the same individual $\endgroup$
    – Yemon Choi
    Commented Jun 24, 2019 at 3:09

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See Example 2.1.6 in this paper.

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