We prove local Lipschitz property of the map which puts in correspondence to each exact $N$-net its Chebyshev center. If dimension of Euclidean or Lobachevsky space is greater than $1$ and the net consists of more than $2$ points we show that this map is not Lipschitz in a neighbourhood of the space of all $2$-nets embedded into the space of $N$-nets endowed with Hausdorff metric.