Limit of $(1+u^{1/n})^n$ as $n\to\infty$ in $p$-adic topology.

Question

I am wondering that if the limit $\lim_{n\to\infty}(1+u^{1/n})^n$ is known to have a closed form.

Also I am wondering what if we take limit in $p$-adic world? In this case, I am mostly interested in the case when $u$ is a $p$-adic unit.