Ramayana was Post-Buddhist and Post-Alexander work !

The period of Buddha was c. 623 - c. 543 B.C.

The ostracised prince Vijayan landed in Sri Lanka in c. 523 BC, assuming he was about 20 years of age at that time. Continued migrations from Bengal and Orissa took place later, at his invitation. So, the people in the East India had become aware of the existence of Lanka by then.

If we study Arthasastra deep, we come to know that Rama or Krishna were not principal deities at that time. Vedic religion accepted worship of Lord Siva only during the period of Swateswatara Upanishad c. 600 BC, the pre-Buddhist era, as the non-idolatrous Brahmins wanted to become priests in the Siva temples. It was only after Buddha's era, they felt the need for their own idolatorus religion. That religion became Vaishnaivism. 

It was during the period of Chanakya, seeds had been sown for that religion. Brahmins hated Siva worship for about 1200 years by then. When they created their new religion, they wanted to propagate it among the masses. Ramayana was born out of that need. One could see all the North-Indian literary works, thereafter, eulogised Vishnu and derided and insulted Siva worshippers. Ramayana was a totally imaginary story and not history. 

All through the Ramayana one could see only the aversion and animosity nurtured by the author towards the opponent of Rama, whose real name was not told at all (which is necessary if that was history). The intention was to poke fun and make the people who hear the story laugh, even when the names are uttered by the story teller. Intense propaganda was there, after the creation of Vaishnavism, through Ramayana.

That it was only a story would become evident from the following facts:

1. Main intention of Ramayana was to denigrate Saivism. That is why Valmiki demonstrated his contempt in giving names to the opponent of Rama and other main characters in the family of that opponent.
2. Ravan means one who is black.
3. Mandothari means one whose belly is like a pot.
4. Soorppanaka means one whose teeth are like winnows.
5. Kumbakarna means one whose ears are like pot.
6. At the same time, Vibeekshana means one who is fearless.
7. Just because Vibeekshana crossed the floor and betrayed his brother and
favoured Rama, he was given a respectable epithet by Valmiki.

How could there be ridiculous names for a great emperor and his brothers and sisters while only one brother was given a honourable name? This would show the extent of animosity nurtured by the Vaishnavites who paid Valmiki to write what they wanted against Saivites.

When Ravan is shown as a great king, learnt in Sama veda and as an expert in music,, why is it that he did not sport for himself and all his family members decent names? This question must be answered to by those who say that Ramayan is an 'Ithihas' (Ithihas means history in Sanskrit and Hindi).

Ramayana was written in the post - Buddhist period, and the people in the East and North were aware of the existence of an island called Lanka.
Valmiki based his story on that premise. It was only a story. it was modelled on Homers Iddyssy when the people came to know about Greek literature after Alexander's invasion. 

Rama told Jabali that a follower of Buddha deserves to be punished the same way as a thief and an unbeliever on par with the follower of Buddha. Therefore, a learned man will not even show his face to one who is suspected to be an atheist”. (Ramayan – Ayodhya Kandam – Sarga 109 – Slogan 35).

These    statements would establish very clearly that Ramayana was post-Buddha era in origin. 

Moreover, R.C. Majumdar himself says, in his article titled 'Evolution of Religio Philosophic Culture in India' – Page 43 – The Cultural Heritage of India, Vol. IV, The Religions, The Ramakrishna Mission, Institute of Culture as under: 

"The first and the last Books of the Ramayana are later additions. The bulk, consisting of Books II--VI, represents Rama as an ideal hero. In Books I and VII, however Rama is made an avatara or incarnation of Vishnu, and the epic poem is transformed into a Vaishnava text. The reference to the Greeks, Parthians, and Sakas show that these Books cannot be earlier than the second century B.C......" - 

The reference to Ramayana at three places in Arthasastra were clear-cut interpolations done by the later day Brahmins. That is visible from the text itself. 

Updated in 2020:

For much more info on this issue, read Pages 38 to 48 of the book 'Right to Priesthood'.