Motivated by a districting problem in multi-vehicle routing, we consider a distributionally robust version of the Euclidean travelling salesman problem in which we compute the worst-case spatial distribution of demand against all distributions whose Wasserstein distance to an observed demand distribution is bounded from above. This constraint allows us to circumvent common overestimation that arises when other procedures are used, such as fixing the center of mass and the covariance matrix of the distribution. Numerical experiments confirm that our new approach is useful when used in a decision support tool for dividing a territory into service districts for a fleet of vehicles when limited data is available.