Wild elk's holding neither breathing device (7)
I believe the answer is:
snorkel
'breathing device' is the definition.
(snorkel is a kind of breathing device)
'wild elk's holding neither' is the wordplay.
'wild' is an anagram indicator.
'holding' is an insertion indicator.
'neither' becomes 'nor' (eg in the phrase 'neither do I').
'elks' anagrammed gives 'skel'.
'skel' enclosing 'nor' is 'SNORKEL'.
(Other definitions for snorkel that I've seen before include "Diver's air-tube" , "Surface swimmer's breathing aid" , "Bent tube for a swimmer's mouth" , "Breathing aid for a swimmer" , "Breathing tube for under-water swimmer" .)