Small communication not permitted to include last name (7)
I believe the answer is:
notelet
'small communication' is the definition.
'notelet' can be an answer for 'communication' (I have seen 'Brief communication' mean 'notelet' so perhaps 'communication' could also mean 'notelet'). I also know that 'small' relates to this answer.
'not permitted to include last name' is the wordplay.
'permitted' becomes 'let' (to let something happen is to permit it).
'to include' indicates putting letters inside.
'last' says to take the final letters.
The last letter of 'name' is 'e'.
'not'+'let'='notlet'
'notlet' enclosing 'e' is 'NOTELET'.
(Other definitions for notelet that I've seen before include "Paper folded" , "means of communication" , "informal communication" , "Decorative letter" , "Short folded letter" .)