I have a small garden, first year. I picked some HUGE cucumbers and cut one open to see if it was ripe. It tasted delicious but then my lips were really sticky. I felt the part of the cucumber that had been cut and it was slowly oozing clear, sticky substance. I put it in the fridge to show my husband. I went to check and the substance had hardened. It was still clear and flaked off in little clear balls.
Do Cucumbers Produce Sap?
Certified GKH Gardening Expert
This is a sign that the plant has bacterial wilt. I am guessing you picked the fruit just as the disease was taking hold and the plant had not wilted yet. Unfortunately, the plant will die and there is not much to be done for it.
This article will help with preventing it next year: