Your question has stimulated great debate within the Christian community. Some have concluded that there is no biblical way to answer the question decisively. Still, there is much that can and should be said.

DEFINE TERMS CAREFULLY

One reason why the idea of demonized or demon-possessed Christians is so controversial is that the terms are imprecise. Does this refer to demons indwelling people in such a way that the people have no choice but to submit and succumb to them, as illustrated in Matthew 17:14-18; Luke 4:33-35; and Luke 8:26-33? Or does being demonized include being actively oppressed or influenced by demonic activity, as illustrated in Job 2:6; 1 Samuel 16:14; and 2 Corinthians 12:7? Distinctions need to be made and definitions clarified.

Since believers are told to resist the devil (1 Peter 5:8,9; James 4:7), believers do encounter some level of demonic work. But does God set limitations that protect believers from the kind of demonic possession or domination that renders them, at least for a time, unable to exercise any freedom of choice in physical or spiritual matters? I assume that this is what you are asking.

DRAW CONCLUSIONS CAUTIOUSLY

There is no explicit Bible statement telling us that believers cannot be demonized in ways that involve the full loss of their will to seek what is good and godly. Nevertheless, since Christians are purchased with Christ's blood (1 Peter 1:18,19) and indwelt by the Holy Spirit (1 Corinthians 3:16; 6:19), and since "the one who is in [believers] is greater than the one who is in the world" (1 John 4:4), a logical conclusion makes it unlikely that a demon can exercise such severe influence on a child of God.

Those who receive saving faith are described as dead to sin, alive in Christ, and under grace; "sin shall not be [their] master" (Romans 6:1-14). It is simply not a dominant characteristic of believers to maintain sinful lifestyles (1 John 1:6,7; 3:7-10). A logical conclusion is that complete demon possession necessitating sin is incompatible with these truths.

Despite all this, there is wisdom in measuring our words and admitting that the Lord, for his purposes, retains the full right to use demonic activity or influence among his people in a way that suits him. Despite the presence of the Holy Spirit within his people, severe demonic influence may still prevail. Recall the "daughter of Abraham" whom "Satan [had] kept bound for eighteen long years" (Luke 13:10-16). When God allows demonic influence, he will limit it, direct it, and use it for his wise and loving purposes. So we need to be cautious in drawing sweeping conclusions.

CONTINUE TO LIVE CONFIDENTLY