Bogeyman in book heading off to trap tardy deer, perhaps (9)

'deer perhaps' is the definition.
(deer are ungulates)

'bogeyman in book heading off to trap tardy' is the wordplay.
'bogeyman in book' becomes 'Fungus' (as in the book Fungus the Bogeyman).
'heading off' means to remove the first letter.
'to trap' is an insertion indicator.
'tardy' becomes 'late' (synonyms).
'fungus' with its initial letter taken off is 'ungus'.
'ungus' going around 'late' is 'UNGULATES'.