That happens because the default bindings for the arrow keys are setup with bind-key -r, specifying that they may be repeated. There are two ways that you can disable this.
First, you can use set-option repeat-time 0 to disable repeating entirely. This will affect all bindings. I find that to be very annoying when resizing panes.
Secondly, you can change the bindings for the arrow keys to use bind-key without the -r option:
bind-key Up select-pane -U
bind-key Down select-pane -D
bind-key Left select-pane -L
bind-key Right select-pane -R