That lady's family in extremely grievous pickles (8)
I believe the answer is:
gherkins
'pickles' is the definition.
(gherkin is a kind of pickle)
'that lady's family in extremely grievous' is the wordplay.
'that lady's' becomes 'her' (belonging to a lady).
'family' becomes 'kin' (synonyms).
'in' indicates putting letters inside.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'grievous' with its middle removed is 'gs'.
'her'+'kin'='herkin'
'herkin' going into 'gs' is 'GHERKINS'.
(Other definitions for gherkins that I've seen before include "They are often pickled" , "Small cucumbers, usually pickled" , "Green fruit related to cucumbers" , "they're usually in a pickle" , "Small pickled cucumbers for her kings" .)