I know that I can exec a date command in my zsh prompt.
However, it shows the old time; to see the current time, I have to hit <return> and get a new prompt with the current time.
Is there a way to configure the zsh prompt to constantly update itself every second?
This would be .... unpleasant in a standard zsh prompt (or bash, or other shells).
I suggest you'd be better off using Gnu Screen.
Screen can have a status line which can show the time. Here's an example screenrc scroll down to "Red Hat Magazine A guide to GNU Screen" to see the sample (i'll reproduce that here) which will, when screen is run, show the current time in the lower right corner of the terminal:
~/.screenrc
hardstatus alwayslastline
hardstatus string '%{= kG}[ %{G}%H %{g}][%= %{=kw}%?%-Lw%?%{r}(%{W}%n*%f%t%?(%u)%?%{r})%{w}%?%+Lw%?%?%= %{g}][%{B}%Y-%m-%d %{W}%c %{g}]'
# Default screens
screen -t shell1 0
screen -t shell2 1