Algebraic independence of $p$-adic numbers

We prove lower bounds for the transcendence degree of fields generated by values of the $p$-adic exponential function. In particular, we estimate the transcendence degree of the field $\mathbb Q(e^{\alpha_1},\dots,e^{\alpha_d})$, where $\alpha_1,\dots,\alpha_d$ are algebraic (over the field of rational numbers) $p$-adic numbers that form a basis of a finite extension of $\mathbb Q$.