Konsole has a selection by keyboard mode. In this mode it is possible to move around the scrollback and select text without the mouse.

Enter and leave this mode by using the keyboard shortcut (Ctrl+Shift+D by default).

Moving the cursor: Arrows, PageUp, PageDown, Home, End.

Moving the cursor vi style: h,j,k,l, to move one character, Ctrl+b,f,u,d for page up/down or half page up/down.

Select text by using Ctrl or Shift with arrows, or by using V to start selection, moving the cursor and then V again to end selection. Shift+V selects whole lines, instead of characters.