Welcome to the community and thanks for the clear repro! 
Yes, this is a known UX quirk: when you use st.date_input inside an @st.dialog, the date picker can keep popping up after selection, especially if the dialog is reopened or rerun—this is due to the dialog auto-focusing the first widget, which causes the calendar popup to reappear. Clicking outside the picker or dialog will stop it, but it’s not ideal for users. This behavior has been confirmed by others and is tracked in Streamlit’s issue tracker (see GitHub issue #11808).
Currently, there isn’t an official fix or built-in workaround in Streamlit 1.56.0. Some users try to shift focus programmatically or add a dummy widget before the date input, but results are mixed. If this bug is a blocker, consider using a custom date picker component (like st-rsuite) or placing the date input outside the dialog for now. If you have a minimal reproducible example or repo, please share it—others may have creative workarounds! Community, feel free to jump in with your solutions or vote on the linked GitHub issue to help prioritize a fix.
Sources: