  Throughout the paper, an analytic field means a non-archimedean complete
real-valued one, and our main objective is to extend to these fields the basic
theory of transcendental extensions. One easily introduces a topological
analogue of the transcendence degree, but, surprisingly, it turns out that it
may behave very badly. For example, a particular case of a theorem of
Matignon-Reversat asserts that if $\mathrm{char}(k)>0$ then
$\widehat{k((t))^a}$ possesses non-invertible continuous $k$-endomorphisms, and
this implies that the topological transcendence degree is not additive in
towers. Nevertheless, we prove that in some aspects the topological
transcendence degree behaves reasonably, and we show by explicit
counter-examples that our positive results are pretty sharp. Applications to
types of points in Berkovich spaces and untilts of
$\widehat{\mathbf{F}_p((t))^a}$ are discussed.
