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Is there an infinite Borel subset of plane which is connected but whose only path connected components are singletons?

I know that a Bernstein set is a non-Borel example of such a set. Thanks!

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A pseudo-arc is an example of a compact connected subset of the plane that is totally path-disconnected.

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I think your assumption should include that the set contains at least two points, otherwise there is a trivial example ...

Eric already mentioned the pseudo-arc, which is of course a perfect answer. Allow me to add a historical discussion, and some more elementary examples.

In addition to the pseudo-arc, any hereditarily indecomposable continuum has your property, by definition. However, there are also many other plane continua (compact and connected subsets of the plane) with this property, including ones that are hereditarily decomposable. I believe Nadler's book on continuum theory has such an example in the exercises, but I do not have it to hand right now.

The first construction of a continuum not containing any curves - a modification of the $\sin(1/x)$-continuum - seems to have been made by Janiczewski in 1912 ("Über die Begriffe Linie und Fläche", in the Proceedings of the 5th ICM). This continuum is hereditarily decomposable, and is probably the first example of a set with the properties you mention.

Since you are not asking for the set to be compact, easier examples are possible. Indeed, any space having an Explosion or dispersion point will trivially have the desired property. In particular, the Knaster-Kuratowski fan is a simple example. Another example is given by the set of endpoints of the Lelek fan, together with its top (which has an explosion point).

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