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We give an example of a nonseparable Banach space      V V       and a function      x x       on [0, 1] with values in the unit sphere of      V V       that is an extreme point of the unit balls of all Bochner      L p {L^p}      -spaces      L p ( λ  , V ) {L^p}(\lambda ,V)      ,      1 &gt; p ⩽  ∞  1 &gt; p \leqslant \infty      ,      λ  \lambda       Lebesgue measure, though none of its values is an extreme point of the unit ball of      V V      . This shows that a characterization of the extremal elements in      L p ( λ  , V ) {L^p}(\lambda ,V)       for separable      V V      , given by J. A. Johnson, does not hold in general.

An extremal vector-valued 𝐿^{𝑝}-function taking no extremal vectors as values