Here's one example. Take any ring $R$ equipped with the following two operations: $$(x,y,z)\mapsto x+y-z$$ $$(x,y,z,w)\mapsto x+(y-z)(w-x)$$
It is easy to see that if you add a constant symbol $0$, then the entire ring structure of $R$ is definable from these two operations. However, for any $a\in R$ the map $x\mapsto x+a$ preserves these operations and thus gives an automorphism. Hence if $R$ is any ring of cardinality $\kappa$ with no nontrivial ring-automorphisms this model will have the desired properties. One such uncountable ring is $\mathbb{R}$; I don't know whether rigid rings of all infinite cardinalities exist but I suspect they do.