Yes. The existence of higher dimensions is entirely probable and even mathematically almost certain. First, let us address one point. There is no reason to discuss spatial dimensions vs temporal dimensions. We do live in a four dimensional world. Einstein showed this and modern day string theory supports this as well (however, I know practically nothing more than that about string theory). The fourth dimension is time. This dimension has unique identities that the others do that have. One, in particular, is that there is no known way to travel the opposite way in time (to go back in time). However with the other three, this is typically not an issue. But any thing is moving through all four dimensions at the same time. (With the exception of light, which in a sense does not move through time....but honestly our current understanding of it's behavior does not allow us to ask that question because it is truthfully unknown). There are, of course, mathematical models that show us this as well. There is a maximum, if you will, about how fast one can travel through space and through time. The faster you travel through space, the slower you travel through time. The faster you travel through time, the slower you travel through space. This is shown in Einstein's Special Relativity Theory.

However, moving on from that spill.. I'll address the question that you are truly interested in. Higher dimensions than our own that are not time.

There is evidence of these dimensions occurring. The Kaluza-Klein Theory predicted that in order for light to travel at speed 'c' is must exist in the fifth dimension. It took a couple of decades before the theory was fully completed (it took a while for them to work out how this stuff worked on the classical level). This theory is one of the initial ground works for string theory.

Although I can't think of anything other than that which might hint towards higher dimensions. If anyone has something to critique on mine or supplement me with more information, please do!