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The Wilcoxon-Mann-Whitney test is a robust competitor of the t test in the univariate setting. For finitedimensional multivariate non-Gaussian data, several extensions of the Wilcoxon-Mann-Whitney test have been shown to outperform Hotelling's T2 test. In this paper, we study a Wilcoxon-Mann-Whitney-type test based on spatial ranks in infinite-dimensional spaces, we investigate its asymptotic properties and compare it with several existing tests. The proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails. We study its performance using real and simulated data.