I fished around in Google scholar and found so many examples that I don't feel like listing any of the links. Nonetheless, a clear picture emerges of an answer that I found a bit surprising: The notation $\mathbb{Z}_p$ for the $p$-adic integers evolved in three separate parts. I should also explain that the real science of etymology is about the evolution of words or notation, not just "when did it first happen".

The subscript notation not only for the $p$-adic integers, but more generally for $p$-adic completions, already appears in several papers in the 1930s and 1940s. For instance Carl Ludwig Siegel says in 1941, "$R$ is the field of rational numbers, $R_p$ the field of $p$-adic numbers, where $p$ denotes any prime number, $R_\infty$ the field of real numbers; moreover $J$ is the ring of integral numbers and $J_p$ the ring of $p$-adic integers". Of course, no one would use this notation today!

The use of $Z$ for the integers has a semi-separate history. I even found an old paper, but more recent than this one by Siegel, that used $Z$ for the integers but $R$ for the $p$-adic integers, with no subscript.

Generally the notation for $p$-adic integers and $p$-adic numbers standardized at $Z_p$ and $Q_p$ in the 1950s. Quite possibly Bourbaki, Algebra, deserves credit for standardizing $Z$ and $Q$ for integers and rationals.

Blackboard bold notation ($\mathbb{Z}$ and $\mathbb{Q}$) came last, at least in print. Despite its name, it's no longer obvious to me that blackboard bold actually first came from blackboards or from typewriters. It's sometimes also credited to Bourbaki, but this seems to be wrong. There is a historical account by Lee Rudolph (in comp.text.tex) that credits certain typewriter models in the 1960s for producing blackboard bold typography for the integers, etc. If that is where it started, then the notation seemed to catch on fairly quickly, although there were holdouts that used ordinary bold for decades after that. (But, before blackboard bold was fashionable, it wasn't even standard to make the set of integers bold $\mathbf{Z}$ instead of just $Z$.)

As an aside, the collision of notation between the $p$-adic integers and the integers mod $p$ is unfortunate. I really prefer to write $\mathbb{Z}/n$ for the integers mod $n$, because it is then written exactly as it reads. Also, partly since it is such a commonly used object, I see no need for extra parentheses, or an extra $\mathbb{Z}$, and certainly just using an $n$ subscript is bad. I'm optimistic that this notation is the way of the future and it would be an interesting separate question in history of notation.

(Sorry, I didn't see the entire string of comments before I wrote all of this. The comments make most of these remarks, but it seems useful to combine them into one historical summary.)

abstractly(using multiplication for the group law notation, as usual in abstract groups) rather than the cyclic group multiplicatively, because many people might want to write that as $\mu_n$. – KConrad Nov 26 '11 at 1:38