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The best quiet pubs in Peckham are along Peckham Rye. Spent plenty of time in these two in my youth - good for drinking/chatting and eating (Herne over Clockhouse). There's also 'The Rye' which is closer to Peckham itself, but not as good IMO. I can make some reccos in Dulwich, too if useful.
https://clockhousepub.com/ (Youngs)
http://thehernetavern.co.uk/Not sure about Elephant, but if you can walk 10 minutes down the road to Borough, there are loads I like and frequented regularly when I used to work on Great Suffolk Street. These are the best of the bunch in my experience. Borough Market has some good spots, but busier.
http://www.thelibertine.co.uk/ (best pizza in London imo)
https://www.thegladpub.co.uk/
http://www.theroebuck.net/
https://lordclydeborough.co.uk/
Anyone got a recommendation for a nice, quiet pub in either Peckham, Elephant and Castle or anywhere in between?