Following discussion in the comments under this question, I have decided to reframe the question and post it here. The main motivation for this is that I am told there is an answer to at least one of these questions and I imagine that will be interesting for me and others.
Question $(1)$:
In a Euclidean space, can there exist a compact subset $K$ with nonempty interior that (deformation) retracts onto $\partial K$?
Question $(1.1)$:
In a Euclidean space, can there exist a bounded nonempty open subset $A$ such that $\overline{A}$ (deformation) retracts onto $\partial A=\overline{A}\setminus A$?
Question $(1.2)$:
Does $(1.1)$ have an answer if we remove the boundedness assumption?
And I suppose question $(2)$ would be, how does the picture change (if at all) if we demand that $A$ or $\mathrm{int}(K)$ are (path) connected?
Yes, I know this has an unusual format but please don't be hasty to close. Consider this a Q&A type post, except I'm not the one answering.
Note that refering to ambient Euclidean space changes the question. In, say, the subspace $[0,\infty)$, you can find some very boring examples such as $[0,1]$ which clearly retracts onto its boundary $\{1\}$... the irritating thing being that technically, in $[0,\infty)$, $\{0\}$ is not a boundary point.
The answer to Q1 (and, hence, Q1.1) is negative, a retraction never exists in this setting, see Theorem 3.20 in
"An introduction to topological degree in Euclidean spaces," by Pierluigi Benevieri, Massimo Furi, Maria Patrizia Pera, and Marco Spadini, https://arxiv.org/pdf/2304.06463.pdf, 2023.
Ultimately, this result is a clever application of the homotopy-invariance of Brouwer's degree. (I am quite sure, one can find some textbook references for this non-retraction theorem, I just do not have time/energy for such a literature search.) Technically speaking, they prove nonexistence of a retraction in the setting of Q1.1. But if $K$ has nonempty interior $A$ and there is a retraction $\bar{A}\to \partial A$, then this retraction extends (by the identity) to the rest of $K$, since the rest of $K$ (the complement to $\bar{A}$ in $K$) is contained in $\partial K$) and $\partial A\subset \partial K$. Conversely, a retraction of $K$ to its boundary will restrict to a retraction of $\bar A$ to its boundary. Thus, Q1 is equivalent to Q1.1.
As for Q1.2: Yes, the answer changes. For instance, $A=[0,\infty)$ obviously retracts to its boundary (the singleton $\{0\}$). In order to have a satisfactory theory for unbounded subsets you have to work in the category of proper maps.