Sources close to the family tell TMZ ... Barb passed away Sunday after being hospitalized last week. We're told she'd been battling non-Hodgkin's lymphoma and other ailments in recent years.

Our sources tell us this touching detail as well ... Barbara apparently held on to life until Sunday, which would've marked the 56th wedding anniversary of her and Don. They'd been married for 52 years prior to his passing in 2017.

Don and Barbara were good old-fashioned Hollywood sweethearts -- and were pretty much partners as Don navigated showbiz in the '60s, '70s and beyond. Barbara served as an EP for a re-release of a comedy show of Don's called 'Don Rickles Live in Concert.' She also appeared alongside him as herself in documentaries and other TV specials over the years.

The story goes that Don met Barbara through his agent, and hit it off.  They were in love and pretty much inseparable. Don always spoke highly of Barbara, once telling Al Roker that he was completely devoted to her and she was his life.

Don's death hit Barbara hard, but she remembered him fondly online ... taking over Don's Twitter account and posting messages on both their wedding and his death anniversary. A couple years back she wrote, "Saturday April 6th will be the 2nd Anniversary of my Don’s passing. It is a very difficult time, but reading your kind Tweets is so helpful. Yes I read them all. Don would have been so pleased to know how much joy & laughter he gave to so many during his lifetime.  RIP ❤️ Barbara."

The couple had two children together, including screenwriter Larry Rickles ... who died in 2011. She's survived by her daughter and grandchildren. Barbara was 84.