At the core of the issue is the lack of buy-in of the procedure.
When doing an audit, if a discrepancy is found between the procedure and what's actually being done, the discrepancy needs to be somehow noted. All that does is point out there is a disconnect. No one is getting slapped, no one should be sent to the guillotine. It's just someone noticing things are not being done the way the document says. Auditors need to be empowered to perform this very simple, and crucial service.
If they are not, the entire audit program could lose credibility... maybe the QMS also.
Once that a discrepancy is acknowledged - and it should be very easy to agree on this basic fact of QMS life - the question becomes one of what to do to align the process document and what is actually going on.
If the document is unclear, is unrealistic, is too strict, or offers no flexibility where some could be allowed, it's possible the document could be changed to match what is really going on.
If, however, activities are being allowed or promoted that do not conform to either the process document, customer requirements or the standard the QMS is built upon, the problem is different, and arguably more serious than an auditor not writing a nonconformity. Section 5 in the standard addresses responsibilities.
Does your audit process say all nonconformances will be addressed as such when found in an audit? If it does, now that you have noticed this issue you should assign a CAR to your own audit process.
As part of the containment action to resolve your own nonconformance, if the previously unassigned nonconformances are important you can issue them and pursue the problem of the process buy-in at that point. If the first level responsible person does not agree with the nonconformance, then the next higher up shold be drawninto the conversation. And so on.
In this way the onus is removed from the auditor - the issue of pressure has been "elevated" to management (you).
As a long term fix, you would need to ask yourself a series of hard questions.
1) Does the auditor have support of the auditee?
2) If so, is the auditor suitable for the work?
3) If so, what kind of remediation is needed?
If #1 is No, then questions #2 and #3 are moot for this immediate issue; the main issue is not clearly one of auditor behavior. Address #1 and then ask the series of questions again.
Understand that performing successful audits requires the ability to perform successful
Crucial Confrontations. (I am not affiliated with the makers, VitalSmarts) It can take some time for a diplomatic person to develop this ability and the confidence to use it, if he or she ever can.
I have a paper titled When Employees Don't Follow Procedures
here.
I hope this helps!