It has been a rumor for years that Lady Gaga and Madonna dislike each other. They aren't best friends, that's for sure. Some of the rumors are completely false. On the other hand, it is true that Madonna has said more than once that she thinks Lady Gaga is a copy cat. She didn't name her name but neither did she need to.
The two have been back and forth on Instagram. They clearly don't have much use for each other. So, yes, it is true they don't like each other. But they aren't arch enemies either as some tabloids say. They just don't care much for each other.
Both women have their own followings. They have both made a great name for themselves. But, they are much alike yet so different and sometimes that makes for good friendships and other times, such as in their case, it does not. It is doubtful that Lady Gaga and Madonna will ever really be friends.