Generally, experts think not. They may, of course, become profoundly depressed and engage in self-harm but suicide requires abstraction of the idea of death:
Good questions. Death — as humans understand it — is an abstraction. Awareness of global reality tells us that
1) All humans (and all life forms) eventually die.
2) Therefore, I, as a human, will die someday.
3) People who die do not ever just come back and resume their former place in this world. That is true of me too.
It isn’t just a massive body of evidence that tells us this; it is abstract reasoning applied to the massive body of evidence.
Here’s a thought experiment: Suppose a pensioner has two dogs and one of them dies. He buries the lost pet in the back garden. Three days later, he comes home through the back way and finds the deceased pet running to greet him, as the other dog barks excitedly in the window. As between the dogs and the pensioner, which will be happy and excited and which will be profoundly shaken by the experience?
The elephants who try to prop up a deceased member of their clan probably don’t have anything like a human sense that their companion is dead. If they did, they would probably not be trying to prop the companion up.Denyse O’Leary, “Are animals capable of committing suicide?” at Mind Matters News
Takehome: Death, seen as the idea of “ending it all,” is an abstraction. To know that you will die one day is to engage in abstract thought. Animals don’t do that. If they did, we’d be in big trouble.
You may also wish to read:
In what ways are cats intelligent? Cats have nearly twice as many neurons as dogs and a bigger and more complex cerebral cortex.
In what ways are dogs intelligent? There is no human counterpart to some types of dog intelligence.
The real reason why only human beings speak. Language is a tool for abstract thinking—a necessary tool for abstraction—and humans are the only animals who think abstractly.