Since you've tagged the question with scrum and scrum-master and daily-scrum, I'm going to assume that you are following Scrum as defined in the Scrum Guide.
Is the Scrum Master attending the Daily Scrum? Although not a required participant, one of the Scrum Master's services is to "facilitate Scrum events as requested or needed". I would expect that a good Scrum Master would occasionally observe the Daily Scrums and, when any of the rules of the event are broken, would speak up and get the event back on track. I would also expect that the Scrum Master would follow up with the Development Team during Sprint Retrospectives to help come up with ideas.
The Scrum Guide also says that one of the responsibilities of the Scrum Master is to ensure that anyone from outside the Development Team that attends the Daily Scrum does not disrupt the meeting. If the Scrum Master is attending the Daily Scrum as an observer or following up with the Development Team on the effectiveness of the Daily Scrum at Sprint Retrospectives, the Scrum Master should learn of any disruptions. Does the team have a 15 minute timebox within which to carry out the Daily Scrum? Does the team find the Product Owner's questions disruptive? If the team is finishing their Daily Scrum in a 15 minute timebox, there is nothing wrong with additional time spent after the event. In fact, the Scrum Guide says that it is common for team members (referring to the entire Scrum Team, not just the Development Team) to "meet immediately after the Daily Scrum for detailed discussions, or to adapt, or replan, the rest of the Sprint's work". If the team does not consider the Product Owner's questions disruptive, or perhaps even considers them helpful in achieving the objectives of the Daily Scrum, why should that change?
It's also important to realize that the Daily Scrum does not need to be in the "three questions" format. The Scrum Guide even says that "some Development Teams will use questions, some will be more discussion based". However, regardless of the format, the purpose is to, on short intervals, inspect how progress is competing toward achieving the Sprint Goals, adjust to increase the likelihood that the Sprint Goals will be met, and raise concerns to the Product Owner if the Sprint Goals are in danger to work to come up with any changes to ensure that a value-adding potentially releasable Increment is available at the end of the Sprint. The Scrum Master can coach the Development Team on different methods, but the Development Team should ultimately drive the execution of the Daily Scrum.
The only point that isn't addressed by the Scrum Master's facilitation of the event and the coaching of the rules of Scrum is the members of the Development Team not knowing what they are going to do next. The priorities of the work should be driven by the Sprint Goal and the attributes of value and order assigned to the Product Backlog Items brought into the Sprint. The Sprint Backlog contains decomposed Product Backlog Items, but all of the decomposed work should be tied to one or more Product Backlog Items (with a value and an order) and maybe have a technical dependency. It should be trivial for the team to determine, once they finish one item from the Sprint Backlog, what the next thing to work on should be.