A job offer for an academic position, which I will not link to maintain anonymity, says "we will evaluate candidates on a rolling basis starting from September 30".
This is the only date present in the job offer.
Is that a "deadline"?
Because I inquired about my application and, while they are still evaluating it, the professor said my application arrived "after the deadline".
It's probably effectively a deadline ("soft deadline") and you've encountered why many job postings and programs just say the deadline is a hard deadline, even though applicants may complain about it (e.g.: why is the deadline so early when they won't make a decision until months later?): it makes it clear to applicants that there is a practical consequence to applying late.
Whether it is or isn't a "deadline" by someone's criteria, it's not like there is a game to play here where you can take the conflict between the website saying "rolling basis past this date" and the professor saying "deadline" and somehow use that conflict to gift you the job. For rolling applications, you've still got the best chance by applying early because the position may be filled by the time a later application is submitted, and they're not necessarily committing to reviewing every single application received before they make a decision. Rather, it is likely that if they consider the search to have failed, only then will they look to see if anyone else suitable has applied in the meantime.
When encountering this sort of thing in the future I would definitely try to submit by the date stated and to be aware that any later submission likely has a lesser chance. That doesn't mean it's not worth submitting later anyways but that will depend on your specific circumstances.
"Academic" covers a lot of ground.
If a Ph.D. candidate position, this may mean that they will evaluate later applications if they haven't been able to fill all the positions and/ or new grant money comes in.
If a faculty position, it may mean that applications from a spectacular candidate will be considered even if very late. OR late applications will be considered if they interview candidates and don't much like any of them, or if a preferred candidate passes for another offer.
Hiring can be messy. On-time applications will get guaranteed review. Late applications may never get looked at.
In some institutions (although apparently not this one) the hiring process can be very bureaucratic and deadlines are absolutely firm.
In my experience "a rolling basis" means "we will keep interviewing candidates until we find one we really like, then we offer that one the job and remove the advertisement". Academic positions can often take a long time to fill, because the academic recruiting would rather have someone who is a near perfect fit than be forced to choose from a set of less than perfect candidates, and is prepared to leave the post vacant for a considerable amount of time until that happens. It's not like in industry where a role typically needs to be filled quickly.
Once that date passes, you run the risk of the position being filled at any time, you could just check the website one day and the post has vanished, or you submit an application but the academic picks a candidate before they have a chance to see it. Applying before that date mitigates that risk, so it is advisable.
Generally no, it isn't a deadline, but they might fill up available slots not too long after that date.
In the specific case, the words "rolling basis" makes it far less likely that it is a deadline.
However, assume that they will have lots of applications by that date and if yours isn't among them, your chances decline, perhaps dramatically.
It could mean that they are not evaluating candidates before that specified date. Then, they start evaluating what they have. If you turn in an application after that date, they may have already started their evaluation process, and so your application may be behind the others that started getting evaluated first. If they are prone to select someone when they find someone who they like and who is fully evaluated, they might be less prone to give later applications as much consideration (maybe you need to look significantly better for them to want to perform all of the later evaluation steps). So, even if it isn't a hard deadline that definitely kicks you out by policy, it may mean that supplying your application after the stated date could be more likely to cause you to have a significant disadvantage.
