I think in untyped lambda calculus you work with expressions, so you take your f function (somehow represented as an expression in untyped lambda calculus), you put that in a fixed point combinator, and the resulting expression is a fixed point of your f function.

It most certainly won't be a representation of a number->number function, but a general element of untyped lambda calculus nevertheless.

It is somehow similar how you take square root of a negative number on the Complex plane. You represent your negative number as a complex number (r \mapsto r + 0i), then you can take a square root of it, but that won't correspond to a representation of a real number.