Smartphone app with Wordfeud-like time constraints In Wordfeud, it's kind of play-and-forget. When your opponent makes a move, you get a notification. Typically I would move once a day while commuting to and from work. A game would typically take a week to finish.

I'm used to playing chess on the eboard FICS client.

However, I haven't found a way to utilise the FICS network to play in this more casual way. It seems that if you disconnect, you defeat. Even if you didn't, you wouldn't get any notifications. I've played on a few websites, where you get an email, but it doesn't really exploit the bells-and-whistles of the smartphone.

So, could you recommend me a client that would do the above?

A smartphone friendly (responsive) website with email notifications is also interesting as a second option.
You're looking for something called Correspondence Chess where you can make moves at your leisure in a time frame you're comfortable with. Chess.com and Chess Time both provide smartphone apps with this capability. Chess.com has many more features and Chess Time only provides correspondence.

iOS and Android for Chess.com 
iOS and Android for Chess Time