I am upgrading from 3.0.12 to 3.1.1 and the automated process fails at "Continue database update"

Updating database to latest stable release

Database type :: mysql4
Previous version :: 3.0.12
Migration already effectively installed (skipped): \phpbb\db\migration\data\v30x\release_3_0_0

The database update has not yet completed.

Continue database update

Clicking on "Continue database update" does nothing. What next?

If I go to the forum, I see

General Error
SQL ERROR [ mysql4 ]

Table 'dbb_cpps.phpbb_ext' doesn't exist [1146]

An sql error occurred while fetching this page. Please contact an administrator if this problem persists.

Ok, so I then followed the alternate instructions for installing, uploading the full 3.1.1 package unmodified (not the config.php file), and left only the images, store and files folders, + config.php, then ran the database update again.

This time it said

Updating database to latest stable release

Database type :: mysql4
Previous version :: 3.0.12
Migration already effectively installed (skipped): \phpbb\db\migration\data\v30x\release_3_0_2_rc1
Migration already effectively installed (skipped): \phpbb\db\migration\data\v30x\release_3_0_2_rc2

The database update has not yet completed.

Continue database update

The first time it said this I hit Continue and it continued for a few seconds, then came back with the abovementioned screen, and this time hitting continue results in a microsecond of activity and then it stops, as per my first message above.

It seems to be failing on the database update, and I cannot think why. I did have a few extra fields (columns) added to some tables for various mods etc .... can't think why else it would fail.