Luk 22:3, 31-32, 3 Then Satan entered Judas, surnamed Iscariot, who was numbered among the twelve…31 And the Lord said, “Simon, Simon! Indeed, Satan has asked for you, that he may sift you as wheat. 32 But I have prayed for you, that your faith should not fail; and when you have returned to Me, strengthen your brethren.” NKJV.
It is impossible for believers to be demon possessed, however believers can be demon oppressed. In contrast between Judas, not a true believer and Peter a true one is this, Luke writes that Satan entered Judas. Judas had no true covering, he was exposed and Satan could do with him whatever he wanted without permission. But Peter and Job (true believers) had Christ’s covering, not to be touched without Satan first seeking limited permission. Satan could not do to them beyond what he was permitted.
Left to Satan whose mission in John 10:10 is to kill, steal and destroy, we would all be destroyed. God may use his oppressions as a means for prunning and sifting as he did with Peter but with the temptation/trials He is always provides a way of escape if we wait on Him. God controls the thermostat.