Promise to drop a pal by the border (6)
I believe the answer is:
pledge
'promise' is the definition.
(I know that pledge is a more specific form of promise)
'to drop a pal by the border' is the wordplay.
'to drop' indicates named letters should be taken away.
'by' means one lot of letters go next to another.
'the border' becomes 'edge' (synonyms).
'pal' with 'a' removed is 'pl'.
'pl'+'edge'='PLEDGE'
(Other definitions for pledge that I've seen before include "Solemn undertaking" , "make toast" , "Bond" , "Give one's word" , "Covenant" .)