While they may look like tentacles, the thin, curly threads that come off the cucumber are actually natural and normal growths on you cucumber plant. These tendrils (not tentacles) should not be removed.
Cucumber plants are vines ad in the wild they are meant to climb up objects in order to take the best advantage of son exposure. The higher up a cucumber plant can climb, the less likely it is that they will need to compete with other plants for sunshine.
In order to do this, cucumber plants have evolved with a system where specially developed leaves are sensitive to touch. These leaves curl around whatever it is that they touch. This allows the plant to literally pull itself up over obstacles for light.
In the modern garden, many times, cucumber plants are grown on the ground without any surrounding supports. Because of this, many people do not realize that a cucumber plant’s natural instinct is to climb. Modern gardeners may not realize that the tendrils on a cucumber are natural.
There is no reason to remove the tendrils from your cucumber plant, even if you do not plan on letting them grow horizontally. Removing the tendrils will cause more harm than good and create a wound that allows bacterial organisms that would harm or kill the cucumber plant.
The best thing to do is to let these tendrils grow naturally. You may even want to consider providing supports for your cucumber plants to grow up. Not only does this provide a more natural environment for your cucumber plants but it will save you some room in your garden.