Important remarks:
a) Update of all Non-UEFI and some UEFI BIOSes:
For a correct function of the Marvell 91xx SATA 6G Controller there are at least 3 different modules needed within the BIOS:
the Marvell 91xx AHCI/RAID ROM module
the Marvell 91xx firmware and
the Marvell 91xx bootloader
 
Some actual AMI UEFI BIOSes have within the CSMCORE file an AIO Marvell 91xx AHCI/RAID ROM module, which contains all 3 components (the uncompressed ROM size is much bigger than 100 KB), but usually the mainboard BIOSes do contain the Marvell 91xx Firmware and the Marvell 91xx Bootloader as separate modules.
If the mainboard BIOS contains these 3 components separately, all of them have to be updated, if the user wants to replace the Marvell 91xx AHCI ROM by a newer version. Usually this should be done by executing the Marvell 91xx firmware tool, which is part of the complete Marvell 91xx Firmware Pack (the actual version 2.2.0.1125b can be downloaded from >here<). 
Unfortunately some mainboard BIOSes do not allow to replace the Marvell 91xx BIOS ROM module by running the firmware installer. Users with such mainboard may have to do it with the appropriate BIOS tool. 
 
Here are the latest 3 module files, which can be flashed into an AMI non-UEFI mainboard BIOS with the MMTool by replacing the old ones:
the MV91xx Bootloader module (in non-UEFI AMI BiOSes usually to find within the GUI line ID E6, the actual version 1.0.1.0002b kann be downloaded from >here<, uncompressed size: 51 KB, unchanged usable with DEV_9123, credits go to ZioGTS for the file).
Important: For a proper function the Marvell Bootloader (=E6) module must contain the correct DeviceID of the on-board Marvell AHCI Controller (as the ROM module itself). That is why the Bootloader module has to be customized before being used for an update. Open it with a hex editor and search for the hex code "4B 1B". It should be there 2 times. Then replace the 2 hex code pairs just behind the "4B 1B" by the correct DeviceID hex codes. Examples: The correct hex code for a Marvell Controller with the DeviceID 9120 has to be "4B 1B 20 91", whereas a DEV_9123 Controller needs "4B 1B 23 91".
the MV91xx Firmware module (in non-UEFI AMI BIOSes usually to find within the GUI line ID E7, the actual version 2.2.0.1125b can be downloaded from >here<, uncompressed size: 200 KB, credits go to ZioGTS for the file)
the MV91xx AHCI ROM module itself (non-UEFI AMI BIOS: it is in one of ID20 lines, the download link to the actual version 1.0.0.1033 is above)
 
b) Update of an AMI UEFI BIOS:
Many AMI UEFI BIOSes just contain 1 single Marvell 91xx PCI ROM module. It includes everything, which is needed for a proper function of the Marvell 91xx AHCI/RAID Controller. This module is situated within the BIOS file named CSMCORE and can easily been updated with the Aptio UEFI MMTool. The download link to the actual complete module containing the Marvell 91xx AHCI/RAID ROM and the additional Marvell 91xx BIOS is above.