Can a being be stuck in an endless cycle of depression and suicides?

You have no permanent self. Because you have no permanent self, you have no permanent characteristics. Because you have no permanent characteristics, you cannot be permanently suicidal.

Can a being be stuck in (etc.)?

Further to @Caoimhghin's answer, I guess that a person -- a "being" -- might even believe so, temporarily.

I think that Buddhism warns that, "there was no view of self that would not lead to suffering" -- and that thoughts like, "this is me", "that was me", and "that will be me", might be described as "unwise attention" or "attending inappropriately" ... and that, conversely, "identity view" (i.e. the "view", the "belief") is one of the "fetters" which people escape as they become more enlightened.

According to DutiyaSāriputta Sutta (SN 55.5), one of the conditions for stream entry is hearing or learning the Dhamma from someone already acquainted with it.

https://suttacentral.net/sn55.5/en/sujato

From that, you could say that get out of the cycle of suffering and narrow-minded ignorance it depends on the change of outer conditions, which may lead that person to the chance meeting a Dhamma follower, or finding casually a book about spirituality/Dhamma/Buddhism/self-help/psychology, or finding someone ready to help such depressed individual.

The mind reacts according to habitual tendencies and to the nature of the presented stimuli (that being internal or external, from the mind or from the outside). If new stimuli is presented, the chances of changing one's world view may increase, specially if the depressed individual is open to seek some help or if the wants to get out of that cycle.

As Ajahn Chah tells us in its Dhamma Talk titled "Living with Dhamma in the World":

"People who drink alcohol sometimes say, "I just can't give it up." Why can't they give it up? Because they don't yet see the liability in it. If they clearly saw the liability of it they wouldn't have to wait to be told to give it up. If you don't see the liability of something that means you also can't see the benefit of giving it up. Your practice becomes fruitless, you are just playing at practice. If you clearly see the liability and the benefit of something you won't have to wait for others to tell you about it. Consider the story of the fisherman who finds something in his fish-trap. He knows something is in there, he can hear it flapping about inside. Thinking it's a fish, he reaches his hand into the trap, only to find a different kind of animal. He can't yet see it, so he's in two minds about it. On one hand it could be an eel, but then again it could be a snake. If he throws it away he may regret it...it could be an eel. On the other hand, if he keeps holding on to it and it turns out to be a snake it may bite him. He's caught in a state of doubt. His desire is so strong he holds on, just in case it's an eel, but the minute he brings it and sees the striped skin he throws it down straight away. He doesn't have to wait for someone to call out, "It's a snake, it's a snake, let go!" The sight of the snake tells him what to do much more clearly than words could do. Why? Because he sees the danger -- snakes can bite! Who has to tell him about it? In the same way, if we practice till we see things as they are we won't meddle with things that are harmful."

(Keep in mind that eels are considered a delicacy in some parts of Thailand).

https://www.urbandharma.org/udharma2/living.html#n13

If the person becomes aware of the dangers and liabilities of living in such conditions, he may want to look for an escape of his/her suffering. In the search for escape, one's perspectives, perceptions, thoughts and worldview become open to the possibility of a better state and a peaceful life. That's when outer help is most useful.

If, on the other hand, that person is not aware of, and open to the possibility of ending that suffering, the chances of that person accepting help decrease drastically. In the fifth chapter of he Dhammapada it is written that:

63. A fool who knows his foolishness is wise at least to that extent, but a fool who thinks himself wise is a fool indeed.

64. Though all his life a fool associates with a wise man, he no more comprehends the Truth than a spoon tastes the flavor of the soup.

65. Though only for a moment a discerning person associates with a wise man, quickly he comprehends the Truth, just as the tongue tastes the flavor of the soup.

https://www.buddhanet.net/e-learning/buddhism/dp05.htm

Have a nice day!