Non-surjective but injective real polynomial functions $\mathbb{R}^n\to \mathbb{R}^n$

Over algebraically closed fields $K$, the Ax–Grothendieck theorem (see also this thread) states that injective polynomial functions $K^n \to K^n$ in $n$ variables are surjective. Is there a simple counterexample for this statement for real polynomial functions i.e. $K=\mathbb{R}$?

The statement of the theorem holds even for $k=\mathbb{R}$. See the article

Białynicki-Birula, A., Rosenlicht, M.: Injective Morphisms of Real Algebraic Varieties.

Evaluating the double limit $\lim_{m \to \infty} \lim_{n \to \infty} \cos^{2m}(n! \pi x)$

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Constructing an $\epsilon$-net of $l_2$ unit ball

Prove that $\frac{(p^{n}-1)(p^{n}-p).....(p^{n}-p^{n-1})}{n!} \in \mathbb{N}$ with $p$ a prime number and $n \in \mathbb{N}$

Show $1+x+(x^2/2!)+ \cdots + (x^n/n!)=0$ has no rational solutions for all $n>1$.

Did Landau prove that there is a prime on $\bigl(x,\frac65x\bigr)$?

Olympiad-style question about functions satisfying condition $f(f(f(n))) = f(n+1) + 1$

Every tree has two leaves. Is my proof ok?

In a real normed linear space if $||x||=||y||$ implies $\lim_{n \to \infty} ||x+ny||-||nx+y||=0$ , then the norm comes from an inner-product space?

Let $p$=prime and $\sqrt{x}+\sqrt{y}<\sqrt{2p}$