A distinction is usually made between positional numeral systems and non-positional.
Let's use Arabic numerals as an example of a positional numeral system. In this kind of system, if we write 100, each digit represents a coefficient in an exponential series. Let's use b to represent the base:
1b2 + 0b1 + 0b0 = 100
Okay, so what about 漢数字? The numbers 一, 十, and 百 each consist of one numeral and each represent different powers of ten, but they do so without respect to position. So in this use of 漢数字, we say they're non-positional. In this system, it makes the most sense to say 十 represents the value ten. It doesn't represent an exponential series like the one above because there's no correspondence between position and degree.
Notice how in the example of Arabic numerals above, I didn't specify a base. It might be decimal, or it might be binary. You can't say the same for 一, 十, and 百. With the numbers represented by these numerals, there is no meaningful definition of b, so it doesn't make sense to talk about what base the numbers are written in.
However, there is a positional use of 漢数字, as well. 漢数字 supplanted 算木 as a positional numeral system in Japan after the 16th Century, and in modern Japan both positional and non-positional uses exist. For example, user1205935 wrote in a comment that a restaurant price may be written
一三八〇円. This number is clearly written using positional numerals:
一b3 + 三b2 + 八b1 + 〇b0 = 一三八〇
We've demonstrated that a base exists! We haven't demonstrated what base it is, and that's the crux of your question. In this case, it's obvious the base is 10. But is it always?
I can't prove it, but I've searched and looked through history books, and I can't find any evidence that it's ever not 10. It seems like there's a very strong preference for Arabic numerals when writing numbers in any base but ten. In fact, I found people joking about what the numeral sequence would be if it did continue, implying it doesn't. (Someone suggested 十土王圭, which I include here for humor value.) So, in the absence of evidence to the contrary, I conclude that the answer is:
No, kanji is not ever used to write values in bases other than decimal.
Two historical notes:
The historical Chinese numeral 十 was once written as a vertical line, and this old form was combined with 一, 二, 三 and the old form of 四 to form single numerals representing the values 11-14. See here and here on Wikipedia. However, I believe these forms had passed out of use before kanji was adopted in Japan.
According to Wikipedia, in the old non-positional system, the juxtaposition 二八 represented the value 16 (二×八） rather than 28. I believe this sort of use is still around to some extent in, for example, 四六時中, which represents 四×六 through juxtaposition.