Does your friend think you should stop being friends because you are fighting frequently? While I think your point from the story is true--that friends don't give up on each other--I also think that even good friends sometimes need a break. The fact that she lives right next door might make that even more true. Perhaps with a little break, your friendship can return to being even stronger.
Another thing to take into consideration from the story...in Sarah, Plain and Tall, there weren't very many people living nearby with whom to form friendships. In our modern world we often have more people around. And for some people--especially introverted people--time alone can be very important.