Could they have been like the Ring and had a “life” of their own?
There is no evidence that the Silmarils were like the Ring or were in any way aware or that they had a built-in master. But this was not needed.
Feanor made the Silmarils using, in part, light from the Two Trees which were the most sacred things in Valinor. And after he made them,
...Varda hallowed the Silmarils, so that thereafter no mortal flesh, nor hands unclean, nor anything of evil will might touch them, but it was scorched and withered;
This happens not through the will of the Silmarils, but as a consequence of the blessings Varda put on them. So when Morgoth stole them, even though he did not touch them directly:
In his right hand Morgoth held close the Silmarils, and though they were locked in a crystal casket, they had begun to burn him, and his hand was clenched in pain;
It didn't go well for Morgoth:
His hands were burned black by the touch of those hallowed jewels, and black they remained ever after; nor was he ever free from the pain of the burning, and the anger of the pain.
For a creature of evil, the Silmarils might as well have been made of red-hot iron.
As Mark correctly points out in his answer, the Silmarils don't have a will of their own, Melkor was burned because he was evil and the Silmarils were blessed by Varda. Tolkien doesn't explain exactly why that happened. However, it seems to be an extreme example of a symbolism used in other places in his writings; people who have fallen into evil being distressed by or feeling physical pain when touching objects created by "good" people.
Maedhros and Maglor
After the War of Wrath at the end of The Silmarillion, Maedhros and Maglor refuse to take the remaining Silmarils back to Valinor (which would have been their redeption). Instead they take the Silmarils for themselves and we read that they are both burned by them.
But the jewel burned the hand of Maedhros in pain unbearable; and he perceived that it was as Eönwë had said, and that his right thereto had become void, and that the oath was vain. And being in anguish and despair he cast himself into a gaping chasm filled with fire, and so ended; and the Silmaril that he bore was taken into the bosom of the Earth.
And it is told of Maglor that he could not endure the pain with which the Silmaril tormented him; and he cast it at last into the Sea, and thereafter he wandered ever upon the shores, singing in pain and regret beside the waves.
The Silmarillion: Quenta Silmarillion, Chapter 24: Of the Voyage of Eärendil and the War of Wrath
Page 262 (Houghton Mifflin Harcourt; 2012 Kindle Edition)
In The Lord of the Rings, we learn that Gollum can't stand anything elvish. This includes lembas and rope made in Lorien, items that are simply made by elves and not even especially blessed.
Shortly after they meet Gollum in the Emyn Muil, Sam ties some of his elvish rope to Gollum's ankle and Gollum feels physical pain at its touch.
He stood over Gollum, while Sam tied the knot. The result surprised them both. Gollum began to scream, a thin, tearing sound, very horrible to hear. He writhed, and tried to get his mouth to his ankle and bite the rope. He kept on screaming.
At last Frodo was convinced that he really was in pain; but it could not be from the knot. He examined it and found that it was not too tight, indeed hardly tight enough. Sam was gentler than his words. ‘What’s the matter with you?’ he said. ‘If you will try to run away, you must be tied; but we don’t wish to hurt you.’
‘It hurts us, it hurts us,’ hissed Gollum. ‘It freezes, it bites! Elves twisted it, curse them! Nasty cruel hobbits! That’s why we tries to escape, of course it is, precious. We guessed they were cruel hobbits. They visits Elves, fierce Elves with bright eyes. Take it off us! It hurts us.’
The Lord of the Rings Book Four, Chapter 1: The Taming of Sméagol
Pages 617-8 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
When Gollum is guiding Frodo and Sam through the marshes, Frodo gives him some lembas which Gollum chokes on.
‘No, we have got no fish,’ said Frodo. ‘We have only got this’ – he held up a wafer of lembas – ‘and water, if the water here is fit to drink.’
‘Yess, yess, nice water,’ said Gollum. ‘Drink it, drink it, while we can! But what is it they’ve got, precious? Is it crunchable? Is it tasty?’
Frodo broke off a portion of a wafer and handed it to him on its leaf-wrapping. Gollum sniffed at the leaf and his face changed: a spasm of disgust came over it, and a hint of his old malice. ‘Sméagol smells it!’ he said. ‘Leaves out of the Elf-country, gah! They stinks. He climbed in those trees, and he couldn’t wash the smell off his hands, my nice hands.’ Dropping the leaf, he took a corner of the lembas and nibbled it. He spat, and a fit of coughing shook him.
‘Ach! No!’ he spluttered. ‘You try to choke poor Sméagol. Dust and ashes, he can’t eat that. He must starve. But Sméagol doesn’t mind. Nice hobbits! Sméagol has promised. He will starve. He can’t eat hobbits’ food. He will starve. Poor thin Sméagol!’
‘I’m sorry,’ said Frodo; ‘but I can’t help you, I’m afraid. I think this food would do you good, if you would try. But perhaps you can’t even try, not yet anyway.’
The Lord of the Rings Book Four, Chapter 2: The Passage of the Marshes
Page 622 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)
The Lord of the Nazgûl
This may be a stretch as it is open to other interpretations. When Merry stabs The Lord of the Nazgûl during the Battle of the Pelennor Fields, we are told that his "sword" is particularly effective because it is the work of the Dúnedain of the north.
So passed the sword of the Barrow-downs, work of Westernesse. But glad would he have been to know its fate who wrought it slowly long ago in the North-kingdom when the Dúnedain were young, and chief among their foes was the dread realm of Angmar and its sorcerer king. No other blade, not though mightier hands had wielded it, would have dealt that foe a wound so bitter, cleaving the undead flesh, breaking the spell that knit his unseen sinews to his will.
The Lord of the Rings Book Five, Chapter 6: The Battle of the Pelennor Fields
Page 844 (Houghton Mifflin Harcourt; Single Volume 50th Anniversary Edition)