The 6th Army was ordered not to surrender ever and it didn't. It was eventually forced to lay down arms after the HQ had been captured and virtually all supplies were exhausted. Hitler not so discretely requested that Paulus kill himself by promoting him to Field Marshall shortly before the collapse of the 6th while at the same time reminding him that no German Field Marshall had ever surrendered. An order Paulus rather famously refused. Interestingly Paulus was a rather devout catholic and believed suicide to be a mortal sin. He expressly forbade it among his men, including the practice of standing up to get shot intentionally by the enemy which he considered the same thing.

In either case, the 6th Army was never out of communications with German command, they had radio telegraphs and communicated with command basically up until they were captured. They were also supplied in part by air through the small airfields at Pitomnik and Gumrak, on some days there were over 100 flights in and out of Stalingrad so written orders could also be delivered in a pinch, at least until these too were lost on January 16th (Pitomnik) and 22nd (Gumrak).

Anyway, Hitler ordered Paulus not to surrender on several occasions, the last I'm aware of was on January 22nd after air supply had ceased. Paulus telegraphed Hitler and requested he be allowed to surrender, Hitler rejected it, also by telegraph. On January 30th the promotion to Field Marshal was issued along with the reminder that no German Field Marshal had ever surrendered, that too was at least by implication an order to fight to the death or committ suicide. Paulus was captured the following morning. The last pocket of German resistance in Stalingrad held out about another day before it too collapsed.