The whole challenge with prompt injection is that if I, an employee with a specific level of access, view ANY untrusted text within the context of the LLM (including pasting text in by hand because I e.g. want it summarized) there is a risk that the untrusted text might include malicious instructions which are then executed on my behalf, taking advantage of my access levels.
The only "access to private data" system that I can think if that's not vulnerable to prompt injection is one where every last token of that private data is known to be free of potential attacks - and where the user of that system has no tools that could be used to introduce new untrusted instructions.