There are no papers in the literature that address the question directly. If you're interested, you can have a look at hep-th/1305.0004 by Fitzpatrick, Kaplan and Poland which addresses a certain large-$D$ limit for conformal four-point functions, but they don't make any definite statements about whether such CFTs exist or not.
Let me make an edit, since some comments mentioned some known examples in $D>6$. There are indeed various examples of interacting CFTs in $D>6$ that obey the axioms of CFT except for unitarity. The idea is that you take some field, let say a scalar field $\phi$, with a kinetic term
L \sim \phi \Box^\alpha \phi
where $\alpha > 1$, either integer of fractional. (You can make rigorous sense of the operator $\Box^\alpha$ with some work.) The scaling dimension of $\phi$ is then $D/2-\alpha < D/2-1$, so the famous unitarity bound $\Delta \geq D/2-1$ is violated. Now for $\alpha$ sufficiently large and $n$ a given integer, you can write down interactions of the form
\delta L = g \phi^n
where $g$ is classically marginal for some $D = D_*$, and then you can repeat the usual RG story (do an expansion in $D = D_* - \epsilon$ dimensions etc.) The same extends to fermions and higher-spin fields. Such theories are interesting but probably not quite what OP is looking for.