The most common and most likely answer to this crossword clue is the 5 plus 3 letter word FIELD DAY
I've seen this crossword clue in Financial Times
F
I
E
L
D
D
A
Y
Buddy explains!
The straight part
We know that "period of great success" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case period of great success = field day.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.