There is a separable Banach space, the Kadec space, for which Ext^n(K,K) is nonzero for every n, so its projective and injective dimensions are infinite.
Cabello S´ anchez and J.M.F
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Homological dimensions of Banach spaces
There is a separable Banach space, the Kadec space, for which Ext^n(K,K) is nonzero for every n, so its projective and injective dimensions are infinite.