Inasmuch as we can trust the ancient sources, Herodotus’ native dialect was Doric, and he may well have been a Carian speaker. As Wikipedia says, we can’t trust the ancient sources anyway: Herodotus
Herodotus wrote his ‘Histories’ in the Ionian dialect, yet he was born in Halicarnassus, originally a Dorian settlement. According to the Suda (an 11th-century encyclopaedia of Byzantium which possibly took its information from traditional accounts), Herodotus learned the Ionian dialect as a boy living on the island of Samos, whither he had fled with his family from the oppressions of Lygdamis, tyrant of Halicarnassus and grandson of Artemisia I of Caria. The Suda also informs us that Herodotus later returned home to lead the revolt that eventually overthrew the tyrant.
]However, thanks to recent discoveries of some inscriptions on Halicarnassus dated to about that time, we now know that the Ionic dialect was used there even in official documents, so there was no need to assume (like the Suda) that he must have learned the dialect elsewhere. Moreover, the fact that the Suda is the only source which we have for the heroic role played by Herodotus, as liberator of his birthplace, is itself a good reason to doubt such a romantic account.
Note that Kos, next door to Halicarnassus, was also Doric speaking; but Hippocrates of Kos also wrote in Ionic. The cultural prestige of Ionia is indeed a likelier explanation, and Wikipedia speculates that “Herodotus appears to have drawn on an Ionian tradition of story-telling, collecting and interpreting the oral histories he chanced upon in his travels.”
There was Doric literature too, but I don’t know of any early Doric literary prose.