Sure, but that would be an object language embedded in the lambda calculus, not the lambda calculus itself. btw, variants of the lambda calculus do exist where

Sure, but that would be an object language embedded in the lambda calculus, not the lambda calculus itself. btw, variants of the lambda calculus do exist where some values (so-called “reference cells”) are object identities. But a value cannot be a lambda abstraction and a reference cell at the same time.