I'm trying to work out a basic example where formal smoothness should fail.
I'm considering $\mathbb{R} \to \mathbb{R}[x,y]/(x^2-y^2)$.
The idea is that not every $\mathbb{R}$-homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}$ should lift to a homomorphism $\mathbb{R}[x,y]/(x^2-y^2) \to \mathbb{R}[\varepsilon]/(\varepsilon^2)$. But I can't see that ever being possible: after all, I can just take the exact same homomorphism, with image $\mathbb{R} \subset \mathbb{R}[\varepsilon]/(\varepsilon^2)$; this gives a valid lift.
Do I need to consider something else than $R = \mathbb{R}[\varepsilon]/(\varepsilon^2)$ with $I = (\varepsilon)$ in order to witness the failure of formal smoothness? I would think that was enough, given that it should fully explain lifting of points to tangent vectors.