Is there a way to keep bananas that are half submerged in milk fresh for at least 26-28 hours?

Assumption here:
The main point where prepping what reads like two servings ("every morning" & "26-28 hours) might come handy, is when you have an uneven number of bananas per serving and storing half a banana is kind of impractical.

Have you considered freezing your fruit? In pieces, so that one banana is cut in an even number of pieces (& can thus be easily divided). Then when it's time to blend your breakfast drink, just add as many pieces of frozen banana as your recipe requires. No need to thaw the fruit, any reasonably strong blender should be able to deal with it.

This would allow you to

• prepare the fruit for multiple servings at once
• eliminate the question of keeping the fruit fresh
• give you bananas that are always just the ripeness you prefer
• lets you make good use of bananas on sale

If still want to go the bananas-in-milk route, it's essential that you refrigerate the whole thing: letting your milk stand at room temperature is not safe. That said, you can even refrigerate bananas that are already blended in milk for a day or two. Food-safety-wise totally fine and while you might see some browning, it's mainly about eye-appeal. So you could blend a two-day ration of bananas in milk.
If your drink contains more ingredients that can't be pre-blended, pour off and refrigerate half of the banana-milkshake, add the daily serving of the other ingredients and whizz to blend for a moment.

Put your blender jar in the refrigerator, covered, if possible. Covering will keep odors from mingling and will prevent your the top of your banana from slightly dehydrating. That said, cover off will be fine too. It might oxidize a bit (turn brown), but that will not really impact your final product.