Rodent following piper finally in childlike story (9)
I believe the answer is:
narrative
'story' is the definition.
('narrative' can be a synonym of 'story')
'rodent following piper finally in childlike' is the wordplay.
'rodent' becomes 'rat' (rat is a kind of rodent).
'following' means one lot of letters go next to another.
'finally' suggests the final letters.
'in' means one lot of letters goes inside another.
'childlike' becomes 'naive' ('naive' can be similar in meaning to 'childlike').
The last letter of 'piper' is 'r'.
'rat' put after 'r' is 'rrat'.
'rrat' put within 'naive' is 'NARRATIVE'.
(Other definitions for narrative that I've seen before include "Chronicle" , "Story recounting events" , "Tale" , "Report" , "Account describing events" .)