This archive contain latest available firmware for HP8562A spectrum analyzer (Rev 910415) for 6 EPROM version of A2 Controller board. Details about history are in firmware-history.pdf.
There is also EEPROM 28C16 copy if necessary.
Based on my view it can become available for users who would like to make an upgrade by themselves on best effort principle because HP8562A is end of support by Agilent from April 2000. 
Using on your own risk.