I don’t really agree with that.
To start with, the fields of every discipline are similar to an evolutionary tree. They branch out from ancestor fields and, even in Physics or Biology, can disappear without leaving any descendants. I think that the argument used that once a philosophical problem is solved it closes the field is simply a question of defining the entire field as that particular problem.
Fields are usually hierarchically organised, with subfields within fields. In addition, even if you can say that all problems in an entire field are solved, the way they were solved can be considered to constitute the field itself as each solution can help in other areas or other problems.
Examples in Physics, my field, are many, but I think that one in philosophy could be more relevant. For instance, I have no problems in identifying a philosopher who is an expert in philosophy of science. Even if this particular problem would be solved, I believe that someone who knew everything about it and how it was solved could also be called such.