An evil spirit from the Lord, means trouble of mind, and/or of body:
An evil spirit troubling the mind: Besides the mentioned King Saul, King David was after the adultery/murder incident also troubled by an evil spirit. While King Saul persisted in his rebellion against God, King David humbled himself and eventually found solace.
Psalm 6 (KJ21) David: "O Lord, rebuke me not in Thine anger, neither chasten me in Thy hot displeasure. Have mercy upon me, O Lord, for I am weak; O Lord, heal me, for my bones are vexed. My soul is also sore vexed, but Thou, O Lord, how long? Return, O Lord, deliver my soul! O save me for Thy mercies’ sake".
An evil spirit troubling the body: The apostle Paul suffered from the workings of an evil spirit. He had asked God to remove this spirit, but got the answer that: "My grace is sufficient for you".
2 Cor 12:7-10 (NIV) "... in order to keep me from becoming conceited, I was given a thorn in my flesh, a messenger of Satan, to torment me. Three times I pleaded with the Lord to take it away from me. But he said to me, “My grace is sufficient for you, for my power is made perfect in weakness.”
An evil spirit troubling both the mind and the body: Job was driven to near insanity when an evil spirit from the Lord destroyed both his family and his health.
Job 13:21,24-26 (NIV) Job: "Withdraw your hand far from me, and stop frightening me with your terrors. Why do you hide your face and consider me your enemy? Will you torment a windblown leaf? Will you chase after dry chaff? For you write down bitter things against me and make me reap the sins of my youth".