According to this question the fly speed can only be used for flying, and otherwise swimming speed is equivalent to half of normal speed. However beholders have a normal speed of 0ft, and flying speed of 20ft.
Can a beholder swim? If so, would its swimming speed be equal to half of its flying speed?
I was trying to Google this question but it doesn't appear to normally be a problem, and couldn't find any answers for it.
This answer is an exact reproduction of Rubiksmoose’s answer to a similar question (speeds adjusted to be appropriate to beholders).
There is no way to calculate the swim speed (as asked in the question) for a creature without one. You can however, calculate how far the creature would be able to move while swimming using other (non-swim) speeds.
The rules for swimming and climbing say:
While climbing or swimming, each foot of movement costs 1 extra foot (2 extra feet in difficult terrain), unless a creature has a climbing or swimming speed. (PHB p. 182)
Note that no speed is specified. In fact, only movement matters at all, not speed. Since you can turn any of your speeds into movement, there is no reason why you would be forced to use one speed over another. Thus, you can use any speed to swim.
This has been confirmed by Jeremy Crawford:
The rule on swimming doesn't specify that you're using your walking speed. You use whatever speed you can to paddle, kick, and otherwise swim.
Since you can use any speeds you have, you can of course use your fly speed. Swimming with a fly speed of [20] would allow you to move [10] feet swimming.
