I cannot find an option to filter tasks by user. Is it possible?

You can filter your tasks by user assignments. One way to do it is to click the funnel icon next to the filter text box, to choose a preset filter.

Click the Assigned to me option to see your assignments, or choose another user’s name from the Assigned to list.

A faster way to get the same results is to type user initials preceded with an @ directly into the filter box: @XY
Your board will highlight the tasks that are assigned to this person.

Aside from being able to show user assignments on one board with filtering, you can also find one’s assignments across all boards with the global search functionality. The difference between filtering and searching is just in their range:
filtering is limited to 1 board, and searching applies to all boards you have access to.

You may also save an applied filter so that it activates the next time you log in.