Exporting hi-res png with small dimensions at exact size

I want to export an illustrator file...

• In PNG format


• Exact dimensions (55x20mm)


• High resolution

72 ppi does not save at high enough standard. 300 ppi makes the dimensions too large.

The end user should never use a file smaller than 55x20mm, so I want to give them the highest quality file possible at that size.

Thanks in advance!