The Lord Jesus healed both physical illnesses and afflictions brought on by evil spirits.
With that model in mind, we can be assured that there is nothing which The Lord God cannot oversee and even employ it for His good purposes, redeeming every evil to declare His glorious Grace. For just as we see Christ's sovereign power at work on the blind, crippled and enslaved during His earthly ministry, proving His authenticity as a spiritual teacher; His saving grace to a modern believer gives no less a testimony to His goodness! And moreover, the Scriptures testify that God works everything together for the good of those that love Him, so whether one is suffering from a physical ailment or an act of spiritual oppression, The Almighty has seen fit to allow such circumstances, and will use them to the betterment of the individual believer who suffers them, as well as for The Kingdom.
That being said, allow us to look for a second to the symptoms associated with the Bipolar condition, which was sometimes called 'Manic Depression', and we see those who struggle with it marked periods of intense and inconsolable grief, suddenly taken by intense and irrational confidence that they can do no wrong - these manic periods not only seeing the subject given to spending far too much money and sexual promiscuity, but also a belief that their ideas and thoughts are higher than everyone around them.
Now if we return to looking at the Lord Jesus Christ, we know that He was 'a man of sorrows, and acquainted with grief' so the Christian who battles with Depression can be assured that their Lord knows their infirmities, and may even come to an understanding that they are being given the honour of sharing in His sufferings for their sanctification, for in His grief, Christ sinned not but sought to do The Father's will.
Yet as to the Mania and that irrational self-confidence, we know that Christ was the only perfect man, so the Christian might be able to temper the extremes of that unwarranted self-belief by recognising they are not so great that they may do as they will, for even God the Son - though His thoughts were most high, and above everyone else - bowed the knee in humble submission to the Will of The Father.
So returning to the initial concern as to whether Bipolar is a demonically influenced illness, well there's no doubt that those in the throes of Mania are a nightmare to deal with, and it might concern those around them that the things a Manic speaks are one minute depraved, while the next lofty and hyper-spiritual… but… the simple truth is that many folks with Bipolar respond quite well to modern medication which curbs much of their excesses, so it's most likely a physical impairment than a spiritual one.