Woman Chloe the Lioness keeps (5)
I believe the answer is:
ethel
'woman' is the definition.
(woman's name)
'chloe the lioness keeps' is the wordplay.
'keeps' says the answer is hidden in the clue.
'ETHEL' is hidden in the letters of 'chloe the lioness'.
(Other definitions for ethel that I've seen before include "- Merman, US actress" , "Dame --- Smyth; William Brown's sister" , "Lethe (anag) -- Barrymore or Merman" , "Perhaps a Barrymore" , "chapel" .)