Let me suggest a reason that you may be experiencing what you are, and it has nothing to do with any lack of ability. But it may, I think, involve intensity of focus. And "brain freeze".
When I'm working on something intensely, I tend to separate myself from all other thoughts and concerns. I dive very deeply into what I'm doing. I tend to do it even when writing here, actually. If someone asks me a question - any question - I freeze up for a bit in order to change context an reestablish a new one before I can answer. Maybe this is all it is with you, also.
If you are thinking deeply about some research issue, large or small, you focus very closely on the problem and various potential solutions, evaluating pros and cons, etc. If someone asks you, right then, to go get the mail, you won't, perhaps, even hear them.
But some people need a bit of time, when asked a question, to establish a context in which that question can be answered. If the question has any complexity, so might the context, and it may take a while to get there. You are "struck dumb" by the question and the impatience of the questioner just makes it worse.
So, here is a thought experiment. Take the same question that "stumped" you when asked and consider how your thought process (and potential success) might have been different if you posed the same question to yourself while sitting quietly with no specific other focus. I suspect that I would be able to work it out fairly quickly, and fill in the empty parts sufficiently well. Perhaps you would also. But the questioner wants an immediate, well formulated answer, just as if you had memorized it.
But memorization is a poor tool in mathematics. You can't (unless you have a very differently organized brain than others) simply memorize all of mathematics, theorems along with proofs. A much better tool for a mathematician is insight that lets you re-assemble the concepts and why they are valuable and true. If you are doing valuable research, I suspect that you have such insight.
But a bit of advice about oral exams. Don't worry too much about having answers instantly on the tip of your tongue. Don't worry too much about wanting to "work out an answer". Doing so aloud can be valuable as it is obvious to the questioner that you aren't stuck, just working. Don't worry too much about making an error, but then, admit that the train of thought isn't going to be productive and say why. This reveals to the questioner that you have a good grasp of fundamentals, and can think productively even if the actual answer eludes you.
I once, taking an oral in Algebraic Topology (not my strong subject), did exactly the above, working out the answer to a question and not getting there. After a bit, I said something like "I'm sorry but this line of thought isn't going to end properly. I can't see the answer, but I can at least explain why this isn't working...". Later I was told that it was a great response. Like a lot of things, it is more important to demonstrate competence than it is to answer every question on demand.
Insight, not memory.