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.)
Before posting this question I did search around a bit (probably inefficiently and certainly quite ineffectively) to see if I could find an email for Elliott Mendelson to ask him directly! But anyway, he picked up my similar query on FOM and very kindly wrote to me:
I was intrigued by your comments about the consistency proof of PA that appeared in the First Edition of my logic book. I omitted it in later editions because I felt that the topic needed a much more thorough treatment than what I had given, a treatment that would require more space than would be appropriate in an introduction to mathematical logic.
I can understand that. Though I think the pointers he gave in that Appendix did spur on quite a few readers to find out more, so I still think it was a Very Good Thing, and it was perhaps a pity to drop it.
[Prof. Mendelson has kindly allowed me to quote him.]
Best Answer
Brian Hayes wrote a column on the volume of the $n$-sphere for American Scientist a couple of years ago, available online here. It includes a bit of history, with bibliography, toward the end, which might be of help here.
Added 4/26/13: Here are a couple of pertinent passages from Brian's article:
"... Sommerville mentions the Swiss mathematician Ludwig Schläfli as a pioneer of n-dimensional geometry. Schläfli’s treatise on the subject, written in the early 1850s, was not published in full until 1901, but an excerpt translated into English by Arthur Cayley appeared in 1858. The first paragraph of that excerpt gives the volume formula for an n-ball, commenting that it was determined “long ago.” An asterisk leads to a footnote citing papers published in 1839 and 1841 by the Belgian mathematician Eugène Catalan."
and
"Not one of these early works pauses to comment on the implications of the formula—the peak at n=5 or the trend toward zero volume in high dimensions. Of the works mentioned by Sommerville, the only one to make these connections is a thesis by Paul Renno Heyl, published by the University of Pennsylvania in 1897."