Show live clock in bash in front of command prompt

I am looking for a way to show the current live time/clock in bash before the command prompt. What I researched so far, there are some possibilities for that. E.g. I know the bash variable PS1 can be customized with t but that only updates the clock every time you press enter.
For example one can put this in ~/.bashrc what I currently have:

PS1='[t] [e]0;u@h: wa]${debian_chroot:+($debian_chroot)}[33[01;32m]u@h[33[00m]:[33[01;34m]w $ [33[00m]'

But that is not what I want – my goal is something like shown here which uses something like following code what I also have in my ~/.bashrc:

while sleep 1;do tput sc;tput cup $(($(tput lines)-1)) 0;printf [`date +%H:%M:%S`];tput rc;done &

This is nearly perfect for me, the only problem with this code is, it works well until I send SIGINT e.g. by pressing control+c. I would like a solution like that which does not stop the clock (or that particular subshell where the clock runs) when pressing control+c. I want to be able to use control+c like usual in bash.

To be clear, I also found a solution for this with zsh but I would like to do that with bash if anyhow possible. And I do not want the clock to be shown elsewhere in the terminal, it should be shown live at the prompt, so I can e.g. even scroll back up the terminal and look at what time/second exactly I issued a command.

Thanks in advance!