Thanks for clarifying. One way you could use the st.download_button would be to convert the file to a .png or .jpg or some other supported filetype.
Otherwise, if you can get a URL for the file, you could do something like this.
geemap seems to recommend the following for downloading the file:
out_dir = os.path.join(os.path.expanduser('~'), 'Downloads')
filename = os.path.join(out_dir, 'landsat.tif')
type or paste code here
You could probably combine the above snippet with some type of button (not the st.download_button) to download the file.