Connect anti-static wrist strap to earth wire?
There are several considerations for your wrist strap. 

A) Safety for you
B) Safety for your fragile components and kit
C) Process, use them correctly
D) Grounded?

A) Your safety

Your wrist strap should include a large amount of current limiting before connecting to a real earth. In practice, this is a 1meg resistor in the strap, and/or where you clip it to, and/or a connector that you plug into an earthed outlet to contact the ground pin.

B) Your kit

There is no point grounding yourself, if the stuff you're working on can float to any potential. Use a conductive sheet, and place all your tools, components, work in progress, on it, and ground the sheet (via a safety resistor) as well.  Metal foil, sheet or a tray will do. In industry they tend to use conductive plastic, which is nicer to work on.

C) Process

Having the right equipment doesn't help if you don't use it correctly. When your PI arrived through the post, it was (obviously) not connected to your grounded sheet. At some point, you have to connect them, and it's at that point a damaging charge transfer could occur.

When you connect them, make sure the point that connects first is a grounded point of the PI, a connector shell for instance. Before you unwrap a component from its conductive bag, or pull it from the conductive foam, touch the bag or foam to bring it to the same potential as you (ground).

D) Grounding?

Once you, you tools, your components, your work in progress are all at the same potential, it doesn't matter whether the whole equipotential group is actually connected to ground or not. In practice, it's a whole lot easier to keep track if it is, and as soon as you use a grounded soldering iron, 'scope  or power supply you have that earth connection, so you may as well start off with it.