Inflexible manifesto ultimately popular within British government (9)
I believe the answer is:
obstinate
'inflexible' is the definition.
(I've seen this before)
'manifesto ultimately popular within british government' is the wordplay.
'ultimately' suggests the final letters.
'popular' becomes 'in' (something that's 'in' is fashionable).
'within' means one lot of letters goes inside another.
'british' becomes 'b' (abbreviation e.g. in 'BBC').
'government' becomes 'state' (the state can mean government authority).
The final letter of 'manifesto' is 'o'.
'b'+'state'='bstate'
'in' inserted inside 'bstate' is 'bstinate'.
'o'+'bstinate'='OBSTINATE'
(Other definitions for obstinate that I've seen before include "Stubborn or unyielding" , "such as a mule" , "Stubbornly persistent" , "Like mules" , "Pig-headed" .)