It all depends what you mean by friendship.
If by friendship you mean dealing justly and compassionately towards them as humans, and cooperating with them in everything that is noble and virtuous, it is not only permissible, but it is precisely our mandate as Muslims--as it has been clearly prescribed in the Quran. Allah orders us ".... and you shall speak unto all people in a kindly way.." (Qur'an: 2: 83);"... And never let your hatred of people who would bar you from the Inviolable House of Worship lead you into the sin of aggression: but rather help one another in furthering virtue and God-consciousness, and do not help one another in furthering evil and enmity; and remain conscious of God: for, behold, God is severe in retribution!" (Qur'an: 5:2).
Furthermore, the categorical imperative of the Qur'an, which is universally valid for all times and places is: "BEHOLD, God enjoins justice, and the doing of good, and generosity towards [one's] fellow-men; and He forbids all that is shameful and all that runs counter to reason, as well as envy; [and] He exhorts you [repeatedly] so that you might bear [all this] in mind." (Qur'an: 16:90).
Therefore, friendship in this sense is not only permissible but is precisely the Islamic imperative for Muslims.
If, on the other hand, by friendship we mean to support unjust causes or condone vices and befriend them for them, it is not allowed. For Muslims are to counsel good and forbid evil and strive to cause a positive change in society. They are to be steadfast in upholding the cause of truth and justice, while remaining united against injustice, wickedness and corruption.
The basic question to ask is who is influencing who and in what way. If by befriending non-Muslims you are exerting an influence for the good that is our duty as Muslims; if, however, by befriending, we are being influenced negatively and being conditioned in habits or actions that are clearly forbidden by Allah, then there is no room in Islam for such a type of friendship. It does not take a scholar to tell us that Islam spread in the world, mainly because of their mercy and compassion rather than through the power of the sword.
Therefore, one must be always on guard against befriending people, unless it is in order to exert a good influence on them and incline them towards godliness, justice, and compassion. Allah reminds us that "On that Day, [erstwhile] friends will be foes unto one another - [all] save the God-conscious." (Qur'an: 43:67).