Without hesitation (9)
(Other definitions for instantly that I've seen before include "Immediately, forthwith" , "With no delay, this very second" , "Without delay or hesitation" .)
'without hesitation' is the definition.
(I've seen this before)
This is the entire clue.
(Other definitions for readily that I've seen before include "Without demur" , "Willingly, quickly" , "Willingly, without reluctance" , "Without difficulty or delay" , "in a willing manner" .)
'without hesitation' is the definition.
(in a flash means immediately)
This is all the clue.
(Other definitions for in a flash that I've seen before include "pictures may be taken" , "Instantly, straightaway" , "How lightning appears" , "Almost instantly" , "Where American detected electricity" .)