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Offline support was added in August last year: https://www.fastmail.com/blog/fastmail-works-offline/
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I literally switched on "Enable offline support", caching "All mail" offline on my iPhone a few months ago. Tons of free space, only using 4GB for offline.

But when my phone is actually offline (on a plane or elevator) it beachballs when trying to find something.

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(I work for Fastmail). That sounds odd – have you contacted our support team so we can look into it? https://www.fastmail.com/support/
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The mobile client now supports viewing emails offline.
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