Is the story of Sita’s banishment present in any scriptures other than Valmiki Ramayana?
Do any puranas mention about Sita's second excile?
Yes , as you have enquired ,Puranas do talk about mother Sita's second excile. Although not in detail as Valmiki Ramayana because it is Itihasa (true history) and Primarily a story of Lord Rama and mother Sita alone. Puranas are mythology (true tails) and talk about various other topics also ,rather than a tail of single god they contain a whole lot of topics related to all the gods and goddesses other narrations and discuss other philosophies of Hinduidm.
For instance take a look at below verse no. 10 of - Skandha 9- Chaptet 11 from Shrimad Bhagvat Purana .Shree Vyasa deva is mentioning a brief account of Shree Rama's life in two chapters 10-11.
He is mentioning in the verse that mother Sita was abandoned (banished) due to fear of people by Shree Ramachandra.
इति लोकाद् बहुमुखाद् दुराराध्यादसंविदः ।
पत्या भीतेन सा त्यक्ता प्राप्ता प्राचेतसाश्रमम् ॥10॥
iti lokād bahu-mukhād durārādhyād asaḿvidaḥ patyā bhītena sā tyaktā prāptā prācetasāśramam
Sukadeva Gosvami said: Men with a poor fund of knowledge and a heinous character speak nonsensically. Fearing such rascals, Lord Ramacandra abandoned His wife, Sitadevi, although she was pregnant. Thus Sitadevi went to the asrama of Valmiki Muni.
अन्तर्वत्न्यागते काले यमौ सा सुषुवे सुतौ ।
कुशो लव इति ख्यातौ तयोश्चक्रे क्रिया मुनिः ॥11॥
antarvatny āgate kāle yamau sā suṣuve sutau
kuśo lava iti khyātau tayoś cakre kriyā muniḥ
When the time came, the pregnant mother Sitadevi gave birth to twin sons, later celebrated as Lava and Kusa. The ritualistic ceremonies for their birth were performed by Valmiki Muni.
Apart from that Padma Purana -Volume 5- PATALAKHANDA - Chapter No. 56 - Rama seeks Bharata's advice. is also mentioning discussion between Shree Rama and Bharata , where Rama orders Bharata to abandon sita in the forest.
From which we came to know about mother sita's second excile.
61-64 O brother , whatever you say is all just. But by my order ,carry out the words which i utter.I know she is purified in fire , is chaste and is honored by the world.(But) I am afraid by public scandal.(Therefor) I (shall) abandon my (wife) Janaki .So taking in your hand sharp and very fiearce sword ,(either) cut off (my) head or abandon (my) wife Janaki in the forest. Hearing these words of Rama ,Bharata with tremor in his body and tears (in his eyes) fell unconcious on the ground.
In the next chapter i.e.57 of Padma Purana Vol.5 we can also find the true reason behind why Rama and Sita were seprated in Sita's second excile
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