The short answer is yes, but a Christian who shows no compassion is an immature one. See “What is a Christian” for more information. There is more to the issue of compassion than simply a mark of whether or not a person is a Christian however.
Compassion is an issue that is hotly debated in the United States. We don’t think of it in such terms, but it appears in terms such as welfare and unemployment extensions. Not to oversimplify the issues because there is more to it than this, but one side says that we should take care of those less fortunate by offering them financial assistance. The other side believes that the best way to help is by forcing someone to work and thus they will take care of themselves and their family. It is the adage of “give a man a fish and he’ll eat for a day. Teach a man to fish and he’ll eat for a lifetime.” Although politics is never this simple and both sides often have selfish motives, in essence both sides are trying to be compassionate but they are displaying it in different ways. (Please no comments about the politics of the issue because I know it is oversimplifying a complex issue.)
Compassion is similar in Christianity. Churches constantly get requests for financial assistance, most often at the last minute and as a last resort. To some, the most compassionate thing to do is to give out money when it is available and requested and if the money is misused, then it is on the conscience of the person who took the money. To others, if the money is needed because of a drug or alcohol problem or because of poor financial management, simply giving money isn’t doing the person a favor. Rather it enables a sinful lifestyle and contributing does more harm than good. There is a fine line to walk when it comes to being compassionate.
When it comes to compassion though, every Christian is going to display a different level that is not related to how mature of a Christian they are. Compassion is a spiritual gift. Some people have a God given gift to be more merciful and compassionate than others. Mercy is pretty much at the bottom of the list of my spiritual gifts. This means that to a person who has a lot of compassion, I’m going to look pretty compassionless. This does not give me or anyone else an excuse to be compassionless, it just means that other people are going to be more compassionate than I am without even trying. In the eyes of someone who has the gift of compassion I may appear to have no compassion. In this instance, it is a matter of perspective. If you have the gift of compassion you are more likely to be frustrated by those who don’t have the gift. The same is true with other gifts. Those who are gifted in evangelism will be frustrated by those who don’t have the same passion for evangelism as they do.