In the clues, the 9 stars represent omitted words (one of which is a dialect variant of another). Those 9 words, along with 12 others which form a set, must be used to complete the 11 unclued entries, which include four two-word phrases and four hyphened words, and are all in Chambers.
Across
1 Government taken in by inhabitants — more than one lied (5)
4 Brook disappointment eventually after Malteser’s crunched (9)
11 Primate, somersaulting, ’ad something to stand on? (6)
12 Private * and expert pilot (7)
14 Prop to walk stiffly (5)
15 Amazon leaves a sign of autumn? (5)
16 Decline after relapse, manic depressive could die ultimately (6)
22 One who may be taken on board by another? (8)
23 Extremely noticeable antique covered in cream (7)
24 1st XI perhaps losing special assistant (4)
25 One fabricating * the wrong way round (4)
29 ‘Chained-up’ alcoholic devices for physicists (8)
32 Creator of bargain basement, say (6)
34 Cover * using helium instead of silver by the ton (5)
35 Very fast, almost very good (5)
39 Tries to interrupt viewer turning over for repeats (9)
40 Marital god greeting soldiers out loud (5)
Down
4 Sportspeople using music cut short? (7)
5 * out of mysterious trance, patients having four spines (10)
6 African country in anger backing Spain falling into *, almost (7)
7 Note car on motorway (5)
8 Mournful Gaelic drunk slips in spilt ale (9)
9 One beginning to fill utility room (4)
13 Dish using soaked bread to * ring (7)
15 Calls on force to have session inside (6)
17 Cheese or * entertaining partners at bridge in peninsula (10)
18 Beryl? Supply me real diamonds (7)
21 In religious festival, Arab benefited from research (6)
28 Determines idiot is holding empty * (7)
30 Servant likely to fail? (6)
31 Info that is being magically...

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