A rigid Urysohn-like metric space

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well-known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $\omega_1$ which has the same extension property as $R$, yet its group of automorphisms is trvial. We also present a similar, although technically more complicated, construction of a complete metric space of density $\omega_1$, having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.