Do non-stable Banach spaces exist?

No, see the following paper of Gowers

https://blms.oxfordjournals.org/content/26/6/523.full.pdf


This is the famous Banach's hyperplane problem that was solved in the negative by W. T. Gowers. There is a whole industry in Banach space theory concerning spaces which have even more peculiar properties (google for hereditrarily indecomposable Banach spaces).

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