The adjective definite is spelled with two '-i-'s (and two '-e-'s, one on each end). The same is true, mutatis mutandis of the adverb definitely. Definitely is often mis-spelled, perhaps because the third vowel tends to be pronounced more as an a than an i; it is an example of the misleading shwa.
There is no '-a-', neither in definite nor definitely.
(This all applies, too, to indefinite and indefinitely.)
The same is true of the whole family of words. Finite[ly] tends not to be mis-spelled, because it is pronounced with two the two '-i-'s as a monophtngal '-i-' ('eye', IPA: /aɪ/) vowels. Infinite[ly], with the same possibility of a shwa in the third syllable, has been written - wrongly - with an '-a-':
This page was suggested by the list of "25 of the most commonly misspelled words in the English language", in the article on "spelling" in Garner, Bryan A., The Oxford Dictionary of American Usage and Style, Oxford University Press, 2000; on line at Oxford Reference Online. Oxford University Press, under licence to Hull University. 18 July 2006. <http://www.oxfordreference.com/views/ENTRY.html?subview=Main&entry=t26.e2017>.
AWE shares the confidence of that article: "Naturally, they're spelled correctly here".
Definitely is also one of the 117 mis-spellings listed as 'Common difficulties' in the section on 'Spelling' within 'Writing' in UEfAP.