Yes, Borazine is aromatic.
According to Hückel’s rules, a compound is aromatic if it satisfies three conditions:
- Complete delocalisation of π-electrons
- Presence of $(4n+2)$ electrons where $n$ can be $1, 2, 3, \dots$.
Borazine is aromatic as there are 3 nitrogen atoms in the structure of borazine that donate two electrons each. Boron doesn't donate any, but it has an empty p-orbital available through which the electrons can delocalize. Hence, in total it has 6 π-electrons. Though the delocalisation is lesser than that in benzene, it still is aromatic.
FunFact: Borazine is also called inorganic benzene because of its resemblance to benzene by structure.