It rather depends on what size unit you're talking about. I'm sure there were some small units cut off for extended periods of time. But, the longest I'm aware of any large unit of paras being cut off was the nine days the British 1st Airborne Division spent at Arnhem.

Operation Market-Garden was a plan designed by Field Marshal Bernard Law Montgomery to end the war quickly by attacking along a narrow front through Holland. Paratroopers would land in a great cordon, stretching from Arnhem in the north to Eindhoven in the south. Their primary objective was to seize a series of vital bridges intact so that British armor could rush north into Germany itself. The American 101st and 82nd Airborne Divisions landed in the south, near Eindhoven and Nijmegen, respectively, while the British were assigned to seize the northernmost bridge (on the Lower Rhine) at Arnhem. The entire plan hinged on speed. The planners knew paras couldn’t hold out for very long against concentrated German counterattacks, so any delays further south would be very bad for the men at Arnhem, who were only expected to hold for two days.

Things began to go very wrong, very quickly for the British paras. A Waffen-SS training battalion west of Arnhem, KG Kraft, blocked the British advance, and at one point attacked and fired on the landing zones themselves. The British sustained heavy casualties from German machine guns and heavy mortars. The special recon squadron that had been assigned to race to the bridge was ambushed and shot up. As additional German reinforcements from the 9th SS Panzer Division reinforced KG Kraft, the British paras found themselves largely unable to break through to Arnhem. Only one battalion, the 2nd Parachute Battalion, was able to skirt the German defenses and reach the city and its vital – and unprotected – bridge.

As the second day opened, the British found themselves in a precarious position. They had to hold the drop zones near Oosterbeek in order to receive the remainder of their troops – only about half of the division had landed on the first day – and glider-borne heavy equipment. At the same time, it was imperative that they break through to John Frost’s 2nd Para Battalion at Arnhem Bridge, which had been surrounded and cut-off during the night. The British were successful in the first task, defeating probing attacks by a grab bag of German units – Dutch SS, students from an NCO training school – but failed, again, to penetrate the continually reinforcing German blocking position. Frost’s men at the bridge continued to hold, mauling a 9th SS reconnaissance battalion on the bridge.

On the third day, things began to fall apart. This was the day on which XXX Corps, the British armored spearhead, had originally been projected to arrive. Owing to a blown bridge at Son, XXX Corps was not even to Nijmegen on the 19th. The 1st Airborne Division mounted a desperate early morning attack on the German blocking position – and was driven back in confusion with very heavy casualties. Reeling, the paras established a hasty defense around Oosterbeek itself. All hope of breaking through to Frost’s men at the bridge was gone. The 2nd Para Battalion continued to hold, but they came under increasing German pressure and, without supplies or reinforcements, the outcome could hardly be in doubt. Their positions collapsed in the early morning hours of 21 September.

The British defenses around Oosterbeek were never broken. But four days of constant fighting without effective resupply gradually wore them down. XXX Corps had reached the Rhine on the fifth day, but with the bridge now held against them, could not do more than provide artillery support to the besieged paras. With their strength almost spent, the remnants of the 1st Airborne Division withdrew across the Rhine on 25 September.