Physics is not competantcompetent to provide an answer to this question, but we can approach it in a reasonable way and point to evidence one way or the other.
The physics that we call 'fundamental' is, at the moment, general relativity and quantum field theory, and combinations thereof. The equation of motion in such theories is deterministic. It is difficult to propose non-deterministic equations of motion without thereby proposing unlikely things such as faster-than-light signalling. So this part of the evidence points to determinism.
Non-determinism is indicated by chaos theory in classical physics, by the unresolved issues surrounding the quantum measurement problem, and chiefly by a third property which I will come to in a moment.
In chaos theory trajectories diverge exponentially, but numbers in science are never precise. This makes it debatable whether or not classical physics is truly deterministic, because the idea of an exact real number in the physical world is sort of a fantasy. As soon as there is a limit to precision, it can get magnified by this exponential sensitivity and soon come to influence macroscopic phenomena. So, within the domain of classical physics, determinism relies on a degree of precision in the physical quantities which may be an unphysical requirement for all we know.
Thus, although chaos theory can be calculated theoretically on a deterministic model, what it shows is that the large-scale behaviour can also be consistent with a non-deterministic model in which the non-deterministic feature is tiny. It follows that all the empirical evidence is consistent with either view. So it is not true to say that the evidence points to determinism. The evidence here is neutral.
In quantum theory the measurement problem is unresolved. There is no consensus among experts that is sufficiently widely accepted to merit the claim that the issue is solved. Among the ways to solve it one can propose that the basic dynamics have a stochastic ingredient. Some aspects of the study of black holes hint at this, but it is an open question.
So far I have written enough to show that we do not know whether physical behaviour is deterministic. Now I will put forward what I think is strong evidence that it is not. This is the behaviour of our human bodies and brains which enables us to be reasonable---that is, to engage in reasoned argument and come to understand mathematics and other things on the basis of their reasonableness. If our brains were totally deterministic then we would think the things we do because the motions of our atoms so dictated. It is hard to see what reasonableness would have to do with it. This is not any kind of proof, but it is a suggestion that our thought processes are able to be influenced by something other than a combination of mere determinism and randomness. If so, then the basic physics of the world is of a kind which would support this. This suggests the world is not deterministic. This, to me, is a reasonable conclusion and it is the one I draw.
Note, this is not about adding some mysterious extra dimension or field or anything like that. It is simply to admit that we have not understood the world in full and a more complete understanding will likely show us that such familiar things as electrons and quarks are following patterns as subtly yet profoundly different from quantum theory as quantum theory is from classical theory.
The connection between free will and the ability to understand is, I should add, not one which all philosophers accept, but it is a philosophically respectable position. Roger Penrose, among others, has supported it by means of a careful presentation which can be found, if I recall rightly, in his book Shadows of the Mind (this is independent of what he or anyone else may think about wavefunction collapse).