In the paper of Ihara
https://projecteuclid.org/euclid.jmsj/1230396372,
he proved that the set of primes which split conpletely in
an infinite unramified extension is finite.
In section 13 of the above paper, he explained about an infinite tamely ramified extension cases.
If every ramification index of primes is finite, the answer is yes.
But I am not sure when the ramification index of some prime is infinite.