I keep mine wrapped in fleece in an unheated greenhouse so would expect it to be OK in your summerhouse. I did lose the leaves one year but the plant regrew from the base so probably hardier than we think.
This depends on a lot of factors - the type of banana, your location, and gardens in cities are warmer than those in rural areas, and of course how low the temperature falls, and how long a particular sub-zero spell lasts. Wrapping up only offers limited protection. If weather is severe and prolonged, then wrapping up and/or moving into a cold greenhouse, will not be good enough.The roots of plants in pots will get colder than roots that are in soil. Bringing a pot into a greenhouse can reduce that effect slightly, but if it's a cold greenhouse and the temperature drops below freezing, then the roots can still freeze.One of the most popular bananas is Musa basjoo. This is relatively hardy and may survive outdoors, with some wrapping up. But keeping this plant in severe conditions is still difficult.The attractive banana with wide leaves, Musa ensete, is not as hardy as basjoo. Ensete also has a very attractive red-leaved variety. The red variety is even less hardy than the green variety. It would be difficult to keep a red ensete without heat.
Folding the leaves up, before wrapping the stem, can give additional protection (but it's not easy).
kaycurtis wrote (see)
..., should the pot be wrapped in bubble wrap
Bubble wrap can help. But it is important to understand that bubble wrap, or fleece, merely act as insulators. They reduce the speed at which the pot gets cold. So if the cold weather is prolonged, and the temperature doesn't rise above freezing during the day, then they will not prevent the pot from freezing.
if its any help, in Cornwall they wrap them in a kind of hessian sacking over the winter if there to big to move such as on roundabouts they seem to be ok come end of winter good luck
Bubble wrapping the pot is fine but never use bubble wrap around the plant itself as this will cause rotting. Use fleece on the plant as this allows the plant to breath. All of my banana's are planted directly into the garden, they all survived the last two bad winters, I always cover the root area with the leaves from the plants and a layer of fleece as well. Its messy to clear up in the spring but the plants survive and that’s the whole point. Before wrapping the stem of the plant remove all of the lower leaves, if possible leave the growing tip only. Then wrap a double layer of fleece from the top to the bottom and tie it with soft string to stop it blowing off. Make sure that the plant is protected right down to the soil level with the double layer of fleece. Check it on a regular basis and all should be well.
I've just purchased a banana plant so will be keeping mine indoors until next year. Mine is a ''Tropicana '' and is only about 15 inches high at the moment but it has a new leaf already I've being trying to find out just how big it will grow as I don't think I will be able to keep it indoors when it gets larger.
Not sure why all my indoor plants are producing leaves this late in the season. I have another new one on my monstera as well.
Musa Basjoo is root hardy to at least -15c or lower. You will lose the leaves to a frost and the pseudostem to anthing below freezing. If you protect the pseudostem with either fleece or straw then you have a 65% chance of keeping it intact. Of course it will depend on the winter temperatures. Ensete are a very different ball game altogether to musa basjoo. They do not like anything below freezing and if the plant experiences a freeze it WILL die. These are best either lifted every year before the first frost and stored dry then restarted in to growth the following spring or simply kept potted and still kept dry over winter. The only thing that will happen when stored dry is the growth will stop and that is it. Make sure the growing medium is 50% perlite and 50% MP compost for good drainage.