What would the rational Urysohn space and the random graph look like if they were uncountable?

We apply the technology developed in the 80s by Avraham, Rubin, and Shelah, to prove that the following is consistent with ZFC: there exists an uncountable, separable metric space X with rational distances, such that every uncountable partial 1-1 function from X to X is an isometry on an uncountable subset. We prove similar results for some other classes of models, for instance graphs. In certain cases we give a (consistent) classification of constructed models.