The image of a map of a small degree is contained in lower dimensional submanifold?

Question

Let $M,N$ be smooth $d$-dimensional manifolds, and let $f:M \to N$ be smooth. Suppose that $\operatorname{rank} df <d$ everywhere on $M$.

Is it true that $f(M)$ is contained in a submanifold of $N$ of dimension smaller than $d$?

(This holds for $d=1$).

Answer 1

Compose the projection $p_1 : \Bbb{R}^2 \to \Bbb{R}$ of the plane onto the $x$-axis with a smooth map $g: \Bbb{R} \to \Bbb{R}^2$ that crosses itself, e.g., take $g$ to be a smooth parametrization of the folium of Descartes. Then the image of $f = g \circ p_1 : \Bbb{R}^2 \to \Bbb{R}^2$ is not contained in any $1$-dimensional submanifold of $\Bbb{R}^2$.