For $\mathbb{Q}_p$, you can have a look at Laurent Berger's notes http://perso.ens-lyon.fr/laurent.berger/CoursM2/Poly2010.pdf, especially theorem 15.3. They are written in French, but guessing from your name, I would not be surprised if you could read French.

Basically, the argument is as follows. Pick an extension $K$ of $\mathbb{Q}_p$ of finite degree d. Let me denote by e the ramification index and by f the degree of the residual extension. We have d=ef. The maximal unramified subextension $L$ is obtained by adjoining a root of unity (of order $p^f-1$). Then you go from $L$ to $K$ by adjoining a root of an Eisenstein polynomial of degree e. Now, the coefficients of such a polynomial lives in the ring of integers of $L$, which is compact. By Krasner's lemma, the extension will not change if you change the Eisenstein polynomial by another one which is close enough. Hence you only have a finite number of extensions and they are generated by elements which are algebraic over $\mathbb{Q}$.

Unfortunately, I am not really familiar with general p-adically closed fields and cannot tell you if the previous argument still works. By the way, do you have any good reference for them?

For $\mathbb{Q}_p$, you can have a look at Laurent Berger's notes http://perso.ens-lyon.fr/laurent.berger/coursM2/Poly2010.pdf, especially theorem 15.3. They are written in French, but guessing from your name, I would not be surprised if you could read French.

Basically, the argument is as follows. Pick an extension $K$ of $\mathbb{Q}_p$ of finite degree d. Let me denote by e the ramification index and by f the degree of the residual extension. We have d=ef. The maximal unramified subextension $L$ is obtained by adjoining a root of unity (of order $p^f-1$). Then you go from $L$ to $K$ by adjoining a root of an Eisenstein polynomial of degree e. Now, the coefficients of such a polynomial lives in the ring of integers of $L$, which is compact. By Krasner's lemma, the extension will not change if you change the Eisenstein polynomial by another one which is close enough. Hence you only have a finite number of extensions and they are generated by elements which are algebraic over $\mathbb{Q}$.

Unfortunately, I am not really familiar with general p-adically closed fields and cannot tell you if the previous argument still works. By the way, do you have any good reference for them?