I'm now 46 years of age. Should I just give up on the possibility of ever having a child?

Weigh your options. At 46 you have a 1/50 risk of producing an infant with down syndrome with additional risks for other complications bringing your risks up past 1/20 for some unexpected outcome. If your a glass half full person, you still have a greater chance of a normal outcome. A session with a genetic counselor could asses any additional issues & point out other options for your quest.
No, but... As long as you're ovulating (haven't gone through menopause) you can get pregnant. But before attempting to do so, you need to have a conversation with your primary care or obgyn doctor because pregnancy after the age of 35 does carry certain risks for you and the baby. This needs to be an informed decision, so be sure and educate yourself properly.