Common emitter amplifier's \$R_C\$ breaks
I assume that IC hits zero,

yes

UCE decreases until the transistor goes into saturation,

yes

but still without collector current. Does this affect the base current?

yes and no. It doesn't affect it much, through hre. So no, if we're dealing with a simple ideal transistor, in which that parameter is zero.

Or Beta?

Yes. Beta is only measurable when the transistor is in the linear mode. You've already established that the transistor is in saturation. With a very simple ideal transistor, you might say that the underlying 'true' beta is constant, so it's not altered by there happening to be zero collector current. However, when you look at the curves for any real transistor, beta varies with collector current, dropping off at very low and very high currents.