The Koch Snowflake has infinite perimeter and finite area. Gabriel's Horn has infinite surface area and finite volume. I'm interested in existence of examples illustrating the converse of the behavior of the snowflake and the horn, except for $n$ dimensions.
For the $n=2$ case (infinite area and finite perimeter), the isoperimetric inequality gives us a nice proof refuting the existence of such a shape, as shown in this question.
Apparently the $n=3$ case (infinite volume and finite surface area) does not possess such an example either, and it's discussed without proof here.
(Note: I'm not interested in degenerate cases like $\mathbb{R}^3$ minus a ball, but I'm not sure how to formally exclude such cases.)
Main question: Is there a way to prove a statement about nonexistence for all $n\geq2$ ?