If a lady who was raped became pregnant through the rape is this lady allowed to have abortion, since it is unbearable for her to keep the baby both mentally and emotionally?

Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

The Fatwa regarding abortion generally has already been given on Islamweb.net so please go back to Fatwa:83872 and 83114.
If the lady is raped with her consent, then causing abortion means committing another sin and a new crime. So, she has to repent sincerely and seek Allah's pardon.
But, if she is raped as you said this means that she is not to blame for the sexual act as the rape implies that it is by force against her wish. However, regarding the abortion, it is prohibited (Haram) if the soul has already been breathed into the fetus. She is to resort to the law enforcement agencies to complain against the aggressor, if possible.
We pray to Allah to help her in this crisis.
Allah knows best.