I'm not sure if you want to find somewhere to find indoors with a reasonable degree of privacy or outdoors. I found bf-ing outside a bit harder than bf-ing indoors initially. In a feeding room (they have them at the Westfield Centre in Derby for example) you can be confident the only people who'll see you will be other people who have kids who should have the sense not to watch (or are too busy looking after their own kids to notice) whereas outside anyone could come along. I'm in Belper so some of these suggestions might be a bit far north for you.
Carsington is lovely but the visitor centre will be heezing over the holidays and you might have difficulty finding a quiet shady picnic spot. Elvaston Castle is a good idea.
At Kedleston Hall you can walk through the park and possibly find a quiet spot in the woods. The walk is buggy-able, we did it last weekend. The tearoom there is also huge, and if you go on a quiet weekday your friend might find a corner where she feels comfortable.
Shining Cliff Woods above Ambergate are a lovely spot for a walk (but you'll want the baby in a sling - not pushchair territory) and you could head to the Bear Inn at Alderwasley afterwards. The pub has lots of small rooms and provided you go on a quiet day you should be fine. Or the garden is huge with big trees so your friend could find somewhere outside.
If you parked at Black Rocks above Cromford and walked west on the High Peak Trail (can use a pushchair) you could probably find some secluded spots around the National Stone Centre.
Lea Gardens are lovely although the cafe might not offer much privacy, the Coach House in Lea might be a good option on a Fri at least.
Downstairs at Le Mistral in Wirksworth the tables are in lots of small rooms so fairly easy to get some privacy and Wirksworth is a nice place to potter around.
If you are ever in Belper the tearoom at I Should Cocoa on Strutt St is very bf-ing friendly, but not especially buggy friendly. If you take the baby in a sling and use one of the tables upstairs you should get a fair degree of privacy and they once gave me free chocolates because I was bf-ing so I love them very much.