Let us say that a topological space $Y$ is a normal absolute extensor if each continuous map $f:Z\to Y$ defined on a closed subspace $Z$ of a normal topological space $X$ extends to a continuous function $\bar f:X\to Y$. By the Tietze Theorem, the real line is a normal absolute extensor and so is the countable product $\mathbb R^\omega$ of real lines and any retract of $\mathbb R^\omega$.

Since each Polish (= separable completely metrizable) space admits a closed embedding into $\mathbb R^\omega$, we conclude that Polish AR's are normal absolute extensors. This fact is explicitely written as Theorem 16.1(d) in the old book ``Theory of Rectarcts'' of S.-T. Hu.

Now it remains to observe that each contractible topological manifold $M$ is a Polish contractible ANR and hence a Polish AR. So, $M$ is a normal absolte extensor.

Let us say that a topological space $Y$ is a normal absolute extensor if each continuous map $f:Z\to Y$ defined on a closed subspace $Z$ of a normal topological space $X$ extends to a continuous function $\bar f:X\to Y$. By the Tietze Theorem, the real line is a normal absolute extensor and so is the countable product $\mathbb R^\omega$ of real lines and any retract of $\mathbb R^\omega$.

Since each Polish (= separable completely metrizable) space admits a closed embedding into $\mathbb R^\omega$, we conclude that Polish AR's are normal absolute extensors. This fact is explicitely written as Theorem 16.1(d) in the old book ``Theory of Rectracts'' of S.-T. Hu.

Now it remains to observe that each contractible topological manifold $M$ is a Polish contractible ANR and hence a Polish AR. So, $M$ is a normal absolute extensor.