According to the last paragraph in Section 3 of the following paper:

http://www.jstor.org/discover/10.2307/2373193?uid=3738736&uid=2&uid=4&sid=21101389828747

the answer is yes. More specifically, they prove that if $a \in \mathbf{Q}_p$ is a non-zero element, algebraic over $\mathbf{Q}$, such that $\left| a \right| < p^{-1/(p-1)}$, then $\exp(a) \in \mathbf{Q}_p$ is transcendental over $\mathbf{Q}$ (ibid.).

According to the last paragraph in Section 3 of the paper "Transcendental numbers in the p-adic domain" by William W. Adams (Amer. J. of Math., Vol. 88, 1966):

http://www.jstor.org/discover/10.2307/2373193?uid=3738736&uid=2&uid=4&sid=21101389828747

the answer is yes. More specifically, they prove that if $a \in \mathbf{Q}_p$ is a non-zero element, algebraic over $\mathbf{Q}$, such that $\left| a \right| < p^{-1/(p-1)}$, then $\exp(a) \in \mathbf{Q}_p$ is transcendental over $\mathbf{Q}$ (ibid.).