Yes, per 510.1c, quoted below in its entirety with emphasis added.
A blocked creature assigns its combat damage to the creatures blocking it. If no creatures are currently blocking it (if, for
example, they were destroyed or removed from combat), it assigns no
combat damage. If exactly one creature is blocking it, it assigns all
its combat damage to that creature. If two or more creatures are
blocking it, it assigns its combat damage to those creatures according
to the damage assignment order announced for it. This may allow the
blocked creature to divide its combat damage. However, it can’t assign
combat damage to a creature that’s blocking it unless, when combat
damage assignments are complete, each creature that precedes that
blocking creature in its order is assigned lethal damage. When
checking for assigned lethal damage, take into account damage already
marked on the creature and damage from other creatures that’s being
assigned during the same combat damage step, but not any abilities or
effects that might change the amount of damage that’s actually dealt.
An amount of damage that’s greater than a creature’s lethal damage may
be assigned to it.
You announce the damage assignment order for the blocking creatures, putting the 2/3 first and the 2/2 second. Assign 5 damage to the 2/3, leaving 0 damage to be assigned to the 2/2.
Since your 5/5 has trample, briefly look at 702.19.b. Since lethal damage hasn't been assigned to each creature that blocked it, you can't assign damage to their control.