First things first, it is possible, though not guaranteed. I did it for my 5KM and 10KM pace, not half and marathon. It comes down to training, and this is a lot of hard training.
Lets start off - a 5:30 mile is a strong pace, compare to your average runner, that is 3:25KMs or 17.56KM/h (10.9Miles/h).
I managed to build my pace over two years. When I started my 1 mile was close to yours, in November my 5 and 10KM pace was faster then my old 1 Mile pace. This was more an accident than a destination.
My journey started by going longer distances. By the time I did a few half marathons and marathons, a 5KM was very simple. I wanted to improve my half and full marathons pace. As such I used the 5KM and 10KM as chances to go hard. I would set the treadmill as high as possible and just go for as far as I could go. This translated to much faster road running. My half and marathon times dramatically improved but no-where near what my 5KM speed is.
Ultimately a 5:30 mile is a good pace to keep for a half marathon-even a 10KM. It would put you in the top 5% for big fun runs.
I say train really hard and eat clean.