The answer to the title question is a simple "No." Not according to the rules, as written.
You need to refer to the section on Actions (p198):
Anything that your character does in a round is an action. It’s
easiest to think of an action as a single thing that you can do in
five to ten seconds... Putting away your bow and pushing a heavy
bookcase to block a door are two actions because each requires a
different train of thought.
The bold element of the quote is important in answering your question. Getting up from the ground and then attacking represents "a different train of thought."
According to Taking Action (p189):
Each character gets one turn each round. On a character’s turn, she
can do one thing — an action... Every action performs a task, and
every task has a difficulty that determines what number a character
must reach or surpass with a die roll to succeed... Most tasks have a
difficulty of 0, which means the character succeeds automatically.
If not resisted or endangered, the action of standing from prone would have a difficulty of 0, but it does represent a complete action. If the prone character had to contend with resistance from the Caffa, per the example, then the difficulty would be higher (at least the level of the Caffa).
Someone with a Special Ability like Divide Your Mind (p36) could manage to rise from prone and attack in the same turn. A GM might make a call on other situations, like a character with the Fast Descriptor (p73) who can move a Short distance and make an action in the same turn.