‘fast’ means ‘completely’ when used with the adjective ‘asleep’
This phrase is C2 in the English Vocabulary Profile.
For example:
Look, Charlie, she‘s fast asleep.
A NOW corpus collocates search for ‘fast asleep’
1 WAS 1032
2 WERE 498
3 WHEN 376
4 WHILE 181
5 BED 114
6 FELL 59
7 BABY 41
8 ROOM 40
9 MOTHER 34
10 WOKE 31