No, it’s temporary protection at best. Lasting immunity happens when your own immune system generates memory cells that can produce antibodies/kill cells infected with a given virus/etc. All that gets passed on through breast milk is the antibodies, not the cells that can make them, so the protection only lasts for as long as the antibodies can stick around. I can’t remember the half-life of antibodies off the top of my head, but it will be in the order of weeks or months at most. After that, you’re on your own.
I look forward to making another post in the “I told you so” thread. 
Anybody got a lactating great grandma?
That’s right. The 1918 flu outbreak actually punched a hole in the middle of the demographic. Turns out that what was most deadly was your own body’s overreaction to the flu (filling your lungs with fluid) so people with suppressed immune systems (old) actually did better.
Fine by me - if there’s a pandemic, I won’t be alive to read it anyway.* If not, then my mere notations that there was a chance that it won’t be the next horrifying disease will just be correct. All I’m saying is there’s no guarantee this’ll be something that’ll wipe out millions of people, and that there’s not a hell of a lot of sense in worrying that it will unless you’re in a position to actually do something about it, like working for the CDC.
- I work with patients with compromised immune systems. In other words, lots of potential carriers.