You can do the mixing in the digital domain at least if the carrier frequency is not too high, but while this is entirely practical at HF, it becomes a much harder problem at UHF as the required sample rates become insane.
Now some of the more interesting parts from folks like analogue, TI & linear tech support an interesting mode where the DAC chip can both upsample and mix with (typically Fs/4 or a multiple thereof) which allows a digital IF to be moved up to a UHF output in the digital domain.
You can of course subsample if the sample and hold bandwidth is sufficient and the bandwidth limits meets the ampling theory limit.
Note that a mixer block in the digital domain is simply a half complex multiplier, and so has none of the non linearities inherent to a diode ring or gilbert cell mixer done in the analogue domain, however there is a limit to the maximum speed that a digital multiplier can operate which is lower then the limit for a analogue mixer.