Continuous surjective map on a path connected subset of $\mathbb R^n$ which induces a covering map on a deformation retract space

Question

Let $A \subseteq B$ be path connected subsets of $\mathbb R^n$, $n\ge 2$. Let $i:A \to B$ be the inclusion map and $r:B \to A$ be a deformation retraction i.e. $r\circ i=id_A$ and $i\circ r$ is homotopic to $id_B$ i.e. $A$ is a deformation retract of $B$.

Let $p: B \to B$ be a surjective continuous function such that $p(A)=A$ and $p|_A:A \to A$ is a covering map. Then is $p:B \to B$ also a covering map ?

Answer 1

No. Let $A = \{0 \}, B = \mathbb R^2 = \mathbb C$. Then $p(z) = z^2$ is a continous surjection, but no covering map.