Hi, although your request for a vet recommendation has many responses, being a pet lover and owner (or better said guardian) of dogs and cats, I offer 2 options that could eventually be helpful.
1) Dr. Alejandro Tirone (Veterinaria Las Marías, Honduras 3731, Phone: 4826-5180). He's treated both my dog and cat on many occasions and spayed my dog. He has a nice way to get around difficult animals such as one of my dogs that doesn't let him touch her and my cat that allowed him to take a loose tooth out.
2) Veterinaria San Marcos. Av. Diaz Velez 4730 (and Otamendi), Parque Centenario. Phone: 4981-4588/4981-1873/4982-0983. It's not in Palermo but it's a 15 min ride in a radio-taxi.
Dr. Castello is the owner and director. You can find him after 5 pm (surgeries in the morning). He is very experienced and probably the best surgeon in BA. He operated on two of my dogs (they passed away at age 19 and 14); in one case of a Sticker's tumour that is a very serious disease that not many vets dare to operate on because of the bleeding.
The veterinary is very well equipped and has vets for every specialty. The eye doctor and the neurologist were also very good assisting my dog when he was very old.
Both are professionaly and morally hightly recommendable. Good luck and hope your pet is well.
The second is not in Palermo but it's open 24 hs and hightly equipped so would recommend keeping the address handy (although hope you never need truly).