Say Paracite, I was wondering but isn't "がんばるな~" more like "You sure work hard" or something like that?
TL;DR It can be, depending on nature of the な. Looking at this again, it's probably closer to 'You sure work hard' than 'hang in there'.
Details and explanation
It's both. Depending on context. It's both 'doing your best' but also persevering; cf. お疲れ or ご苦労 which is for when you have already completed the work. You can also compare it to 頑張って which really only has the meaning of perseverance.
For 'You sure work hard', 働き者 would be ideal for this (since it does hard worker).
This is assuming the な isn't a shortened ない (which would not be an extended な in any case), but just the mild intensifier. To me, 頑張るな~ is implying that Shioi is encouraging, rather than commenting on the Admiral. (You wouldn't get 頑張るね, you'd have 頑張ってね, which would be very much 'hang in there').
You could say that she is saying that he is a hard worker for continuing to work, and that she is not saying it in a negative light, which results to the same thing, in the end.
However, if it was 頑張る人 or something, then yeah, 100% 'you work hard'.
Looking at my dictionaries and whatnot, 頑張る is almost always translated in sentences as 'do your best' in most situations, with 'work hard' in only a few, select instances. (often with a negative - don't work so hard').
So maybe, in this case, something more along the lines of 'do your best' would be better than 'hang in there', but they have similar meanings, given the context.
And now I'm second guessing myself - if the tone of the extended な is up, it's encouragement (and not very very long), but if it's down/flat, then it's likely a comment, and thus better translated in such a way.
cf
(提督、)がんばるな~ (提督は)がんばる(人)な~
Where the first is encouragement, and the second is commentary. (Bracketed words are elided); hence in the first, it's talking about how the Admiral should continue to put effort into the tasks at hand, while the second is talking about how the Admiral does put effort into the tasks.
Now I'm just confusing myself. On the basis that the な does have a long line on it (and Shioi's expression, though it's Maryu-yu-san, so everyone's a bit odd looking anyway), I think that I'm leaning more towards comment now.
I see. I guess you're right. I didn't study japanese much, all I'm based is from what I hear in anime. And I was thinking twice and maybe the right word for "you sure work hard" was "がんばったな~" and I think I mistook these two.