= Solution
Write $\Gamma=\operatorname{Gal}(K_\infty/K)\cong\mathbb Z_p$. Its only finite subgroup is trivial. In particular, a real place cannot acquire complex inertia in this <Zp-extension>: the possible nontrivial inertia at a real place has order two. Thus every infinite place splits in the tower.
Suppose that no finite prime ramified. Every finite layer would then be an abelian <everywhere unramified extension of number fields>, including splitting at real places. Every layer would lie in the ordinary <Hilbert class field> of $K$, a finite extension. Their degrees are unbounded, a contradiction. Therefore \b[some finite prime must ramify].
Let $v$ have residue characteristic $\ell\ne p$. Apply <local class field theory> to the corresponding <decomposition group>, a closed subgroup of $\Gamma$. The image of the unit group $\mathcal O_{K_v}^\times$ is the <inertia group>. Its first principal-unit subgroup is a <pro-l group>, so its continuous image in the <pro-p group> $\Gamma$ is trivial. The residue-unit quotient is the finite group $\mathbb F_{q_v}^\times$. Its image is finite, and is therefore also trivial in the torsion-free group $\mathbb Z_p$. The entire <inertia group> is trivial. Hence
$$
\boxed{v\text{ ramifies in }K_\infty/K\ \Longrightarrow\ v\mid p.}
$$
Equivalently, the <tame ramification> relation with a Frobenius element would force a tame inertia generator to satisfy $\tau^{q_v-1}=1$, which is impossible nontrivially in $\mathbb Z_p$.
For the final assertion, the <cyclotomic Zp-extension> of $K$ is $K\mathbb Q[p^\infty]$. Fix $v\mid p$ and put $E=K_v$, a finite extension of $\mathbb Q_p$. Let $C_n$ be the completion of $\mathbb Q[p^n]$ at its unique prime over $p$. This is a <totally ramified extension> of $\mathbb Q_p$ of degree $p^n$. In a compatible local algebraic closure,
$$
e(EC_n/E)
=\frac{e(EC_n/\mathbb Q_p)}{e(E/\mathbb Q_p)}
\geq\frac{p^n}{e(E/\mathbb Q_p)}.
$$
Here the inequality follows from the tower through $C_n$. The right side is unbounded. These composita occur among the completions of the cyclotomic tower over $K$, so $v$ has unbounded <ramification index>, in particular nontrivial <inertia group>. Thus
$$
\boxed{\text{Every prime }v\mid p\text{ ramifies in the cyclotomic Zp-extension.}}
$$
This argument allows an arbitrary finite intersection between $K$ and the rational cyclotomic tower; it does not assume disjointness.
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