The short answer is yes, if no government is formed, we could see new elections as early as mid-January, or as late as end of March, depending on the path Netanyahu and president Rivlin take.
Today, Rivlin gave Netanyahu the mandate to try and form a government. He now has 28 days to complete the task, and the president may, if he so chooses, extend it with up to 14 days more. If he fails, Rivlin may opt to give the mandate to Gantz for another 28 days with no extension. If that fails, the Knesset has 21 days to form a majority around any member and ask the president to let him try to form a coalition. If that fails, elections are held within 90 days. This brings us to elections on March 25th 2020.
But more likely, this will end up much earlier, one way or the other. At his acceptance speech, Netanyahu said he will give his attempt a few days (commenters speculate it will be about a week), and if he sees no progress in the negotiations with Kahol Lavan, he will return the mandate to the president (unlike in April, when he passed a law to dissolve the Knesset before he had to return the mandate). At this point it's quite possible that the president won't waste more time giving Gantz a chance, and move on to give the Knesset a chance to find a candidate. If that fails, we will see elections on January 21st.
After such elections, if held, it's the same process all over again.
Another idea that started to float in recent days, for example by former Labour party chairman Avi Gabay, is that if no government is formed, pass a temporary law for quick direct elections to the prime minister only, which will give a clear direct answer to the question of the identity of the PM, and end this saga.