Valve has trademarked the name Half-Life 3 in Europe.
The mark was filed on September 29.
Take from that what you will.
Update: Half-Life 3 might be in development yet. Valve’s User Picker—internal software Valve uses for project management and not meant to be public—has added group categories for “Half-Life 3” and “Half-Life 3 Core.”
Filtering these groups using the “In Group” drop down shows a total of 46 staff members on the “Half-Life 3” project and 10 staff members on the “Half-Life 3 Core” project. You can try it yourself.