What is the daily work of a UX designer like?
It depends on several factors, primarily the stage of the project(s) being worked on and the type and size of company.
I have found that the typical design process traditionally has four stages: Analysis and Research, Conceptual Design, Design Testing and Refinement, and Design Documentation.
In the first phase you may be interviewing (or helping to interview) users and/or stakeholders, or reviewing legacy designs and industry research. This is a very focused effort to understand the problem space and identify pain-points and inputs into design. Lots of sticky-notes and conversation involved.
During Conceptual Design you sketch and prototype and iterate and then you sketch some more. You come up with candidate designs. Lots of whiteboarding and pencil-on-paper work here (and if you want to "flesh out" a particular design direction you create more detailed designs using design tools like Omnigraffle). Part of this phase also involves defining the "high level" design direction and design principles if you don't already have those set up.
During Design Testing and Refinement you test your design with users, in whatever form you have the designs (could be paper sketches, could be more fleshed out prototypes). This allows you to see what works and what doesn't. This takes quite a bit of time, and you iterate and refine the design based on feedback to improve it. In addition to test facilitation, lots of the same type of activity in the first two phases takes place here (communicating, designing, problem solving).
Design Documentation involves writing up the detailed design documentation and findings from the user testing. This involves writing and creating lots of UI screens. Lots of writing.
What type of company influences how much time is spent in each phase (some companies may give design documentation a high priority, others may focus more on design testing). A consulting company may not spend a lot of time doing design documentation, for example. The size of the company may influence the size of the design team, so you may have a larger team of more specialized UX designers with people who focus on one or two key disciplines (like user research). These people will spend most of their time doing the activities that occur in a particular phase for multiple projects.