I'd have to say Yes. Kenya was a British colony until Dec 12 1963, when it gained independence. Google tells me this much (but it's still subject to variables) ....
Birthright Status (1949–1963): Anyone born in the colony between Jan 1, 1949, and Dec 11, 1963, was generally a Citizen of the UK & Colonies (CUKC).
Independence Act 1963: On Dec 12, 1963, most people acquired Kenyan citizenship, and automatically lost their CUKC status.
The Exceptions: People were allowed to retain British status (becoming British Overseas Citizens or, in some cases, full British Citizens) if they, their father, or their paternal grandfather was born in the United Kingdom or a place that remained a colony.
Disparity in Citizenship: This, in practice, favoured white settlers with direct, recent connections to Britain, while many Asian and Black Kenyan residents found themselves in a precarious position regarding their right to reside in Britain or Kenya.
SWMBO initially married a bloke in the late 1960's, who was Kenyan born - of mixed descent. His father was British and Anglo-Saxon. But his mother was born in Nairobi, of Dutch-Lebanese/African parentage, and her parents came from the Seychelles.
But SWMBO's daughter (my stepdaughter) found she was entitled to a British passport, because her dad, and both his parents, came to Australia in the early 1950's, on British passports.