Is there a compact $n$-dimensional manifold $M$ or, more generaly, a compact $n$-dimensional topological space $M$ with the following property?

"For every continuous map $f:M \to \mathbb{R}^{n}$ there are points $a,b,c \in M $ with $f(a)=f(b)=f(c)$."

This is motivated by the following obvious consequence of the Borsuk-Ulam theorem:

"For every continuous map $f:S^n \to \mathbb{R}^n$ there are points $a, b \in S^n$ with $f(a)=f(b)$."