The necks in the 60s varied, so:
Do you know what year exactly? What is the nut width of your Fernandes?
Edit: the answer to your OP question is yes there are some reissues that get pretty close to a 60s neck, but the necks vary by which year the reissue is modeled after.
Exactly! Until we know the size/profile of that late 60s neck it's impossible to give any kind of meaningful advice. There is really nothing special about 60s Fender necks in general, except that they were very inconsistent and that many of them are in a rather bad conditions these days. It's all about the dimensions.
Most of the responses so far, where people are just recommending their own favorite basses (in some cases not even based on the neck...) or even worse, basses that they've only read about online, are quite frankly useless and should be ignored.
If we know the nut width, the thickness (front to back) at the 1st and 12th fret and a rough description of the neck profile (is it for example a classic C or leaning more towards a U...?) we might be able to find a bass with a similar neck, or a Fender compatible replacement neck. It's impossible to give you good advice without this information.
But even with all this information, you won't really know until you hold the bass in your own hands. Even rather subtle differences can make a HUGE difference. For example, it may not seem like a big deal if the thickness at the 1st fret varies by 1 mm, but it can significantly change the "feel" of the neck.
The one thing we DO know is the fretboard radius, which is 7..25" on late 60s Fenders. Pretty much all P-basses sold today, with the exception of Fender Vintera and Fender American Vintage series are 9.5" radius. But both the Vintera and the AV have the wider "vintage style" nut width, similar to what Fender referred to as a C-sized neck back in the 60s (not to be confused with a C profile neck). The neck on your friend's bass might be a C-size, but it might also be a B-size which has a similar nut width to "modern" P-basses.
The neck "sizes" (nut widths) Fender used back then were, from thin to wide:
A: Jazz Bass
B: Similar to "modern" P-basses
C: Similar to current "vintage reissue" Fender P-basses
D: Even wider (extremely rare, probably meant from players who made the transition from double bass)
This means that a P-bass could even be ordered with an A (J-bass) sized neck size back then, but based on the vintage P-basses I've seen over the years this didn't seem to happen very often. 60s P-basses with original J-bass necks seem to be very rare.
This is not to be confused with neck PROFILES, which on a late 60s P-bass is probably a C but might also be closer to a U,