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My wife is on my Tier 1 dependent visa and was originally issued around Nov 2011 and she has had two extensions since then, her latest renewal was on 6th Jan 2017 and she has been in UK continuously since Oct 2016 i.e., more than 3 years now. Is she is eligible to apply for British Citizenship without ILR as she is 3 years and I am a British Citizen. Any suggestions / advice are welcome and thanks in advance.

Ilr is mandatory before she can apply for citizenship. There are no exceptions.

She would have to apply for ILR first no matter what. Once she has ILR she can directly apply for british citizenship without waiting for 1 year as your are British citizen.
first check her eligibility for ILR.