It is known that Krishna, Narada etc., are married Naishtika Brahmacharis.
Is rama a Naishtika Brahmachari?
No he wasn't because he had married.
There are four orders of life in Hinduism viz: Brahmacharya (studentship), Grihasta (householder) Vanaprashtha (retired) and Sannyasa (renunciate).
A Naishthika Brahmachari is one who lives in the first order of life perpetually i.e. till his death. He does not marry.
Quoting the necessary verses from this answer:
A Brahmna, whose tongue, generative organ, belly and hands, have all been controlled, should, resorting to Sannyasa (renunciation), lead a celibate life near the same preceptor, as long as he lives ; in his absence near his (preceptor's) son ; [in the latter' s absence,] near his disciple or in his family. Neither marriage nor [absolute] renunciation is laid down for the Naishthika* (13,14)
(Harita Smriti verses)