The question of whether human beings have a non-physical component is a long-standing debate in philosophy. It's sometimes known as the mind-body problem. Are all of the features that we associate with mind (such as consciousness, personal identity, conscience, and so forth) real, or are they simply the effects of the functioning of our brain?
Materialists of course believe that our mind has no reality or existence apart from our brain. They sometimes point to the fact that people with brain damage often undergo dramatic changes in their behavior and personality to show that these things are simply products of brain function. This objection, however, does not address the question. Nobody denies that there is a connection between the mind and the brain. The question is whether the mind just is a product of the brain's activity, or if there is something more to it. As Dinesh D'Souza has pointed out, "When I bash my radio, the sound stops. Does that mean the radio is creating the sound waves? Isn't it more reasonable to say the radio is simply the instrument or conduit that makes it possible for us to hear those sound waves? . . . The brain may well be the necessary vehicle for mental activitity. It does not follow that brains and minds are identical."
But in my opinion the biggest problem for a materialist conception of the mind is that of free will. If our minds are simply the product of our brain chemistry, then free will is an illusion. Our thoughts and actions are determined by the laws of physics and chemistry, and we have no control over them. Indeed, there is no "we" to control them, because personal identity is also an illusion if materialism is true. "We" simply are the product of our changing brain states, and those states change because of purely physical causes.
While some materialist philosophers of mind just simply say "okay, free will is an illusion," in my experience most people aren't willing to do that (of course, whether they are unwilling or simply predetermined is an open question!). Most people believe in their own ability to choose. In fact, I would go so far as to say that even those who deny free will still live and act as though free will was real. They deny it in theory, but not in practice.
One big problem with jettisoning free will is the problem of making a coherent case for moral responsibility. That is, if we don't have free will, how can we be morally responsible for our actions? Some philosophers have tried to overcome this hurdle without much success. Morality and moral responsibility depends upon our ability to make moral choices.
In an interesting study, subjects were given passages from a book to read about consciousness. One group was given a passage that said that free will was an illusion. The other group had a passage that said nothing about free will. All the participants were then given a math test in which they were told how they could cheat on the test without being caught, but then were told not to. The group that read the passage denying free will were more likely to cheat, and the more strongly they denied free will based on a survey they had filled out, the more likely they were to cheat. You can read more about that here. This study would seem to empirically demonstrate that there is a connection between belief in free will and moral behavior.
If we have free will, then materialism is false. Contrary to what modern materialists tell us, we do have a non-physical mind. The fact that our mind may survive our physical death should be enough reason for anyone to seriously consider what their destiny after death will be.