Whether the algorithm is polynomial or not with input size which is not polynomial
This depends on how precisely the words are meant.

In computer science, we often lack perfect (or even good) information about the behavior of algorithms. As a result, we tend to approximate or create upper bounds and talk about those instead of actual behavior. In most contexts, when you say “the algorithm requires superpolynomial space” what you mean is that in the worst case, we cannot prove that the algorithm uses less than superpolynomial space.

This is very different from the literal interpretation of your words, where an algorithm requires superpolynomial space if there exists an input for which it use superpolynomial space.

So why does this matter? Writing $k$ units of space generally requires at least $k$ units of time in the contexts where we are interested in doing space analysis. Therefore if the algorithm in fact requires superpolynomial space, it must require superpolynomial time. Additionally, if we cannot prove it doesn’t require superpolynomial space, then it follows that we cannot prove that it doesn’t require superpolynomial time. However, it is possible for us to not be able to prove that it doesn’t require superpolynomial space and in fact requires polynomial time.

Additionally, it’s possible that in your context writing $k$ units of space can be done in much less than $k$ time. There do exist computation models where space is much cheaper than time in this sense, but in those contexts we rarely care about space complexity, because it’s so cheap.

So, the answer in most contexts is no, but it’s very important to think carefully about what your words mean.