The title says it all. I know a lot of tourist destinations are closed in the winter.
Are there things that specially open in winter so you can do them when holidaying there at that time of year?
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There is always walking, even in winter you can go out and walk.
What kind of walks depends on the actual weather and your condition and preparations. There are people who walk the hardest trails in the dept of winter. Most people will need to take it a lot easier.
Ambleside is (almost) on lake level, which makes it easier to find low level walks in cold weather.
There are a few sights in Ambleside, I am now not sure which are open in winter, I will edit this answer later when I have found opening times.
Early in the winter, there will be Christmas markets in several towns and villages in the Lake district. Unlike those in the big cities, in the Lake district they are mostly just one weekend or even one day events. So you will need to look when there is one, and which is near enough.
Here is one list, a random one from internet. You will see that some of them have already been, even on the 6th of December.
If the winter is snowy, you can ski in the lake district. Here is a link to one site, note, they do not have ski hire nor teaching. There may be other ski locations as well, google might be helpful there.
And as high as the hills in the Lake district are, snow is much more likely than in the rest of England.