The halting problem is computable with literally-infinite time. But, to be precise, what this means is that a hypercomputer could determine whether a (nonhyper)computer halts; in a universe containing hypercomputers, we would not be very interested in that, and we’d be asking for something that determines whether a given hypercomputer halts (or something like that; I haven’t given much thought to what corresponds to “halting” for any given model of hypercomputation...) which would be impossible for the same sort of reasons as the ordinary halting problem is impossible for ordinary computers.
But I think it’s only fair to describe this by saying “the halting problem is impossible even with infinite computational resources” if you acknowledge that then “the halting problem” isn’t a single problem, it’s a thing that varies according to what computational resources you’ve got, getting harder when you have more resources to throw at it.
The halting problem is computable with literally-infinite time. But, to be precise, what this means is that a hypercomputer could determine whether a (nonhyper)computer halts; in a universe containing hypercomputers, we would not be very interested in that, and we’d be asking for something that determines whether a given hypercomputer halts (or something like that; I haven’t given much thought to what corresponds to “halting” for any given model of hypercomputation...) which would be impossible for the same sort of reasons as the ordinary halting problem is impossible for ordinary computers.
But I think it’s only fair to describe this by saying “the halting problem is impossible even with infinite computational resources” if you acknowledge that then “the halting problem” isn’t a single problem, it’s a thing that varies according to what computational resources you’ve got, getting harder when you have more resources to throw at it.