I was wondering if anyone might have a non-trivial example of a formally etale algebra over a field of characteristic 0 which is not ind-etale (i.e. a union of etale extensions).

For some motivation, over fields of characteristic $p$ there are many such: just take the limit or colimit perfection of any $\mathbf{F}_p$-algebra: e.g. $\mathbf{F}_p[t^{1/p^{\infty}}].$

I have not been able to construct similar examples over $\mathbf{Q}$ say and am wondering if anyone here has come across such an example. Perhaps there are formal reasons why it is not possible?