Influencing yes, but not embodying. That Satan was there at that time is a given. But if he had taken over Peters body and mind, which Satan is NOT allowed to do without peters permission, the story would read differently from what it does.
From another standpoint. Lying spirts do not need permission to lie in order to cause a vision to appear as lying wonders (wonder )the faith principle.
Christ as the Son of man experienced a lying wonder brought by the father of lies, the god of this world. He did not ask permission but waited for an opportunity. Not eating can affect how a person thinks and therefore believes exercising working their faith out.
Some call it out of the body experience .But Christ did not move one inch from where he was standing either time. Both times the father of lies took him up as when he was shown all the kingdoms of this world and the glory of them .Over and over as it is written, cast out the lies. In that way it’s not what goes into a man’s thoughts but what comes out.
And when the tempter came to him, he said, If thou be the Son of God, command that these stones be made bread. But he answered and said, It is written, Man shall not live by bread alone, but by every word that proceedeth out of the mouth of God.Then the devil taketh him up into the holy city, and setteth him on a pinnacle of the temple,And saith unto him, If thou be the Son of God, cast thyself down: for it is written, He shall give his angels charge concerning thee: and in their hands they shall bear thee up, lest at any time thou dash thy foot against a stone.Jesus said unto him, It is written again, Thou shalt not tempt the Lord thy God.Again, the devil taketh him up into an exceeding high mountain, and sheweth him all the kingdoms of the world, and the glory of them;And saith unto him, All these things will I give thee, if thou wilt fall down and worship me.Then saith Jesus unto him, Get thee hence, Satan: for it is written, Thou shalt worship the Lord thy God, and him only shalt thou serve. Mat 4:3