In this paper we deal with the asymptotic behaviour of sample maxima of Lp-norm asymptotically spherical random vectors. If the distribution function of the associated random radius of such random vectors is in the Gumbel of the Weibull max-domain of attraction we show that the normalised sample maxima has asymptotic independent components converging in distribution to a random vector with unit Gumbel or Weibull components. When the associated random radius has distribution function in the Fréchet max-domain we show that the sample maxima has asymptotic dependent components.