This is a very interesting question and my (short) answer is NO, protein domains are not "indivisible".
However, it depends on what you mean by "protein domain" and "indivisible".
According to Wikipedia: A protein domain is a conserved part of a given protein sequence and (tertiary) structure that can evolve, function, and exist independently of the rest of the protein chain. Each domain forms a compact three-dimensional structure and often can be independently stable and folded.
This is only an partial extract but seems like a good starting point.
Also critical to your question is the word "indivisible".
For many, if not all protein domains, it would be possible to change, add or subtract multiple amino acids without altering the structure. Most domains vary in length, for example Immunoglobulin-like domains contain about 70-110 amino acids.
So how much of a division can you introduce and still retain function.
You are asking about partial protein domains - again the question is how much of the domain is retained? If it's possible to estimate/determine how much is lost then one might be able to determine the likelihood of retaining function. As an estimate it may be possible to retain some function having lost say 40% of amino acids.
Many domains have binding or enzymatic activity. There are often critical residues that are involved in this activity. Thus a lot of activity can be lost with the loss of key residues.
Does this help at all?
Best wishes, Paul