According to WP:
[The Church-Turing thesis] states that a function on the natural numbers can be calculated by an effective method if and only if it is computable by a Turing machine.
Important in this definition is that the functions are on the natural numbers, and this is important, because computability and Turing machines are related to arithmetization very strongly, which relates to the topics like arithmetization of analysis (encyclopediaofmath.org) and reverse mathematics. In fact, if you've done coursework in language and automata, you'll know that alphabets and strings constructed from them are all sequences, and mathematically, a sequences is a mapping of any symbols to the naturals. Thus, the naturals, languages, and Turing machines which serves as a mathematical model for computability are all interrelated.
Suppose there is a physical process that allows for the calculation of functions not computable by any Turing machine... Would the existence of such a process refute the Church-Turing Thesis?
SO, the answer to your question is a qualified yes, and it is easy to see because in the definition cited the biconditional operator (IFF) is used, and therefore where you have a system modeled by a Turing machine, then you have a system of physical computation (SEP). Deterministic TMs and probabilistic TMs are great for determining deductive proofs and stochastic systems, resp. Non-deterministic TMs as an even larger class, which likely excludes quantum computer systems in theory since they can be restated in terms of deterministic TMs (QMSE) by simply reformulating complexity requirements, still seems to encompass every known physical system. This correspondence can be taken as a metaphysical necessity (SEP) such as a view held by Piccinini:
The success of the Church–Turing thesis prompted variations of the thesis to be proposed. For example, the physical Church–Turing thesis states: "All physically computable functions are Turing-computable."
So, the class of formalisms related to TMs seems to encompass all traditional notions of computation (as differentiated from pancomputationalist computation) and if suddenly, a physical process were discovered, say in some new theory of quantum gravity, that would, based on the definition invalidate the equivalence between the LHS (functions on naturals) and RHS (TM) by showing they're not equivalent, that there is some physical system that a Turing Machine of some form can't model.
BUT, it's hard to imagine what a physical system that can't be arithmetized in some form would be. Arithmetic is built into our very systems of mathematical physics that undergird our theories of the physical. So, from a practical perspective, Piccinini's position is intuitively very secure, and it may be indication that the computer as a physical artifact, which encompasses both hardware and software, are evidence of some form of property dualism such as the position Chalmer's has put forth to square the magical relation between the physical (hardware) and the mental (software).