Note : The question isn't super specific on that in contrary to its title, so I'll assume that you have Assistant turned off or don't want it. I don't think there would be ads otherwise
OP stated that turning off auto update on the play store fixed his issue, which seems very random and probably unrelated to me. There's a sure way to disable these notifications though, and it's starting to be a known trick for people who don't want Google Assistant at all :
By changing your device's language to a less known regional variant of it, you'll make Google think that Assistant isn't available for you and your country, thus preventing it to be offered to you anymore by those "ads" or by long home button press.
For example, if you live in the U.S, and speak English, your device's primary language probably is "English (United States)". Changing it to "English (U.S Virgin Islands)" will make any mentions of Google Assistant disappear from your phone. In my case, my device's language is French (France), so I changed it to French (Swiss) to get rid of Assistant.
Keep in mind : Assistant may someday be available to all English countries so this solution isn't completely future-proof.
Self-Note : I made screenshots for a How-To part but I couldn't post them due to my low rep on this stackexchange. I'll probably update the answer when I'll be able to post these.