Yes. Any person, including a believer, can have an evil spirit in their body and/or their soul, which is the mind, emotions & will of a person. However, the "spirit" of every human being is reserved exclusively for God to come into & dwell in.
Jesus said, "that which is born of the Spirit is (human) spirit.." (John 3:6) showing that when a person is born again [John 3:7] it is the person's own "spirit" that is birthed by God's Spirit.
But, even a Spirit-filled believer can still have enemy spirits that are influencing & even attached to their body [like cancer or some other physical afflliction] or influencing or attached to their mind, emotions or will, hindering advancement in Christ.
Importantly, the Greek word in the New Testament for our English translation of "demon-possessed" has the actual definition as being "demonized". So, whether a person has an evil mindset or is completely taken over by demons who control that person's "will" (Mark 5:1) , then according to Scripture that person is "demonized".
Scripturally, there are definitely various levels of demons & also various levels of being 'demonized' ranging from a person being oppressed with evil spiritual activity to the most extreme case where the person is completely taken over by demons which I call 'full-blown' demon-possession, as in the case of the demoniac found in the Gospel of Mark Chapter 5. [ Rob Conrad ]
Can even a believer have demons?