Can You Kill The Landowner?

Judging from all evidence, it would seem that the Landowner cannot be killed. This is because of a choice by the developers no doubt, and may serve to help you to realize how fragile and delicate your survival is in Sir, You Are Being Hunted.

A video of someone attempting to kill the Landowner: Attempted Landowner Kill

He was not successful, and used '38 Rifle Shells, 28 Shotgun Shells and 1 Dynamite' or something to that effect.

It is, however, possible that there is a way to kill him that relates to the plot. I have not played the game, so I could not answer from personal experience.