Let $f:X \to S$ be a smooth morphism and $S$ the spectrum of a discrete valuation ring. If the generic fiber of $f$ is rationally (chain) connected then is the special fiber of $f$ also rationally (chain) connected? We know this to be true if $S$ is over a field of characteristic zero, due to Kollar, Miyaoka and Mori. If not true in general, is there any known special cases when this holds true? Any hint/reference will be most welcome.



Deformation invariance of rational connectedness in positive/mixed characteristic
In characteristic $ p>0 $, RCC and SRC (separable rational connectedness) do differ (see Janos Kollar's (Rational curves) book: V.5.19, I know that Janos Kollar refers to this even though I do not have the book.). In **Higher Dimensional Varieties and Rational Points** pg. 41 and pg. 43, Carolina Araujo and Janos Kollar show that SRC is deformation invariant. 


Namely they refer to Kollar IV.3.11 and Kollar, Miyaoka and Mori to show that the existence of a very free curve is an open condition (condition 6) and the existence of a chain of rational curves connecting any two points (condition 1+) is a closed condition. I do not have access to these sources, so you will have to check that the proof that condition 6 is an open condition does not use properties that only hold for varieties over a field of characteristic zero.


Assuming that there are no problems with condition 6 being an open condition, then since $ \operatorname{Spec}(R) $ is connected, if $ \mathcal{X} $ is a scheme over $ \operatorname{Spec}(R) $ and the general fibre of $ \mathcal{X} $ is SRC, then the special fibre is also SRC.


I don't know about whether RC is deformation invariant in positive characteristic, so I cannot answer that part of the question per se.


