Is my 6mth old allergic to dairy products?

I've started weaning my 6 mth old. I gave her yoghurt for the first time and she developed a red rash on the skin around her mouth almost instantly. She didn't even eat a mouthful. I tried again the next day and the same thing happened. I waited 4 days and I tried a cow milk protein based formula in a bottle for her. The rash appeared instantly. I want to wean her off milk, so what can I give instead? And should I try dairy products again later? Or can it lead to something more serious than a rash?