None of the existing governments will remain, but there will be governments under the King of Kings and Lord of Lord's. That statement itself shows there will be but there are others to say that there will be governments under Christ...
Eze 37:24 And David my servant shall be king over them; and they all shall have one shepherd: they shall also walk in my judgments, and observe my statutes, and do them.
Eze 37:25 And they shall dwell in the land that I have given unto Jacob my servant, wherein your fathers have dwelt; and they shall dwell therein, even they, and their children, and their children's children for ever: and my servant David shall be their prince for ever.
David will be king over all Israel. There will be twelve rulers under him...
Mat 19:28 And Jesus said unto them, Verily I say unto you, That ye which have followed me, in the regeneration when the Son of man shall sit in the throne of his glory, ye also shall sit upon twelve thrones, judging the twelve tribes of Israel.
There will be rulers of cities...
Luk 19:16 Then came the first, saying, Lord, thy pound hath gained ten pounds.
Luk 19:17 And he said unto him, Well, thou good servant: because thou hast been faithful in a very little, have thou authority over ten cities.
Luk 19:18 And the second came, saying, Lord, thy pound hath gained five pounds.
Luk 19:19 And he said likewise to him, Be thou also over five cities.
Rulership is the reward of the saved...
Rev 5:10 And hast made us unto our God kings and priests: and we shall reign on the earth.
There is much more inferred, but these are directly spelled out.