Can an undocumented immigrant go back to their country by plane with no problems?

You should have no problems.
But you should consult with an Immigration attorney if you ever plan to enter the USA in the future.
Your departure will trigger the ten year unlawful presence bar based on the facts presented.

When it comes to immigration laws, you can leave when ever you want, unless you have some pending legal matter, the problem will always be coming back. I know it is out of the scope of your question but keep in mind, if you have been here for so many years dnd want to leave it is ok, what if you don't fit in Peru and want to go back. Once you are there, there won't be an easy way to come back.

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You can leave the US with no issues whatsoever. Nothing to worry about.

Carl Shusterman (former INS Trial Attorney, 1976-82) has 40+ years of experience practicing immigration law. He is a Certified Specialist in Immigration Law who has testified before the U.S. Senate Immigration Subcommittee as an expert witness. He was featured in the February 2018 issue of SuperLawyers magazine. His response to your question is general in nature, as not all the facts are known to him. You should retain an attorney experienced in immigration law to review all the facts in your case in order to receive advice specific to your case. Mr. Shusterman's statement above does not create an attorney/client relationship.

Generally, undocumented do go back to their country by plane with no problems.

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