Places for animal lovers near South London?

So far I've been to London Zoo (too expensive to go regularly) and the Cat Cafe near Spitalfield Market (nice, but a few too many humans competing for the cats' attention).

Anyone got any other recommendations?

South London isn't a must, but I'm in Lambeth so ideally should be reachable from there. I'm looking for basically anything where there's a likelihood of seeing animals, whether that's wildlife, or zoo animals, or farm animals, etc. Interacting with them would be a bonus.

Also I'm a student (moved here in October, hence why I don't know what's around) so ideally it either needs to be relatively affordable OR have a student discount.

Any ideas? It's been over a month since I got to go home for Christmas and visit my parent's cats, and I am suffering withdrawals.