The classical spatial median is not affine-equivariant, which often turns out to be an unfavourable property. In this paper, the asymptotic properties of an affine-equivariant modification of the spatial median are investigated. It is shown that under some weak regularity conditions, the modified spatial median computed by means of the sample norming matrix is asymptotically equivalent to the one computed by means of the population norming matrix, which yields its asymptotic normality. A consistent estimate of the asymptotic covariance matrix of the modified spatial median is also presented. These results are implemented in a scheme, where the sample norm is determined by means of the sample Dümbgen scatter matrix. The results are utilized in the construction of affine-invariant test statistics for testing the multi-sample hypothesis of equality of location parameters. The performance of the proposed tests is demonstrated through a simulation study.