I'm not normally one for posting about religious venues but I've heard good things about this.
All Saint's Centre Norwich
Opposite John Lewis
It describes itself as a "centre of Christian hospitality"
It's a nice drop in centre - It's a cafe but you there's no obligation to buy anything -
"A warm place to sit - stay as long as you like"
A quiet lounge
Free Christian book lending library
"We welcome anyone to our Centre whether in need of information, a rest, a chat, somewhere to meet friends, to leave luggage"
Open Monday to Saturday 10.00 am to 4.30 pm