Recent macOS versions have altered the behavior of the Quick Look process, which has had an inadvertent effect specifically on the handling of Photoshop files stored on network shares.
In this case, Apple's Quick Look function holds onto a file and prevents changes to it, when Finder's default icon preview option is allowed (this shows the PSD image next to the file, rather than the generic PS icon). The process will eventually time out on its own after Finder is closed, but this isn't very useful in normal operations.
Right-click in Finder, go to "Show View Options", and uncheck the "Show icon preview" box.
This needs to be done separately for every Finder mode: column, list, gallery, etc.
This also needs to be done on every workstation that is going to be looking in that directory. If one user is browsing the directory with icon prevew turned on while another is trying to save, the save will fail.
This will not take effect immediately if the directory was initially opened with icons on. The quicklookd process must be killed (or the workstation needs to be rebooted).