Salam,

Thank you for trusting me with your question. Some heritage Muslims will tell the convert to Islam that they must cut their ties with all non-Muslims because of the following verse in the Quran. {[…] whoever makes friends with them [non-Muslims], these are wrong-doers}.

I am sure these brothers and sister in faith who convey this idea to converts, have the best intentions in mind. But it seems the idea of not having ties with non-Muslim is not exactly true. The context of the verse is important:

{Allah does not forbid you respecting those who have not made war against you on account of (your) religion, and have not driven you forth from your homes, that you show them kindness and deal with them justly; surely Allah loves the doers of justice.

Allah only forbids you respecting those who made war upon you on account of (your) religion, and drove you forth from your homes and backed up (others) in your expulsion, that you make friends (and ally) with them, and whoever makes friends with them, these are wrong-doers.} (Al-Mumtahanah 60:8-9)

It is quite clear that Islam teaches us to keep ties of kith and kin and to treat people with kindness. This is especially true of family ties. Allah says in a Hadith Qudsi:

Allah, the exalted, says: I am the Most Merciful, I created Ar-rahim (the womb) and I cut out a name for it from my name, so whoever keeps good ties with it, I will keep good ties with him, and whoever cuts it off, I will cut him and finish him off. (Imam Ahmad, Abu Dawud and At-Tirmidhi)

Here we understand the ties that are created from the womb (the family ties) are very important. 

Furthermore, in the following verse we understand Allah’s reason for making us all different, and it is not so that we can break ties:

{O you people We have created you from a male and a female and made you into nations and tribes that you may know each other. The best among you is the most pious} (Al-Hujurat 49:13).

Dr. Said Ismail Seeni says about the above verse that:

Islam did not come to break the ties between human beings, to spread hatred and enmity, or to encourage bloodshed. Nor did it come to eliminate the freedom of choice in this life, which is secured by God. Indeed, even Satan was granted the freedom to choose between obeying and disobeying his Lord, he misused this gift and chose to disobey.

So, rest assured that you should absolutely keep family ties and keep your non-Muslim friends. But you should also be sure not to allow these ties to lead you to the haram. 

I hope this helps. 

Salam.