The point space $*$ is a smooth $0$-manifold.
I cannot determine if one should think that whenever $M$ is a smooth manifold, any map $M\to *$ is a smooth map of manifolds. I am not sure how to check the condition, since it would require me to write down a matrix with $0$ rows for the Jacobian (or check that all zero of the partial derivatives exist)? Does it hold vacuuously?