Unfortunately, no. The boost you can steal from a powered flyby is only of the same order as the escape velocity of the massive object. For the sun, this is 617 km/s at its surface, or 0.002 c. The term you're interested in is "Oberth effect".
To get relativistic speeds on the table, you'd need exotic, compact objects whose escape velocities are relativistic. I.e. white dwarfs, neutron stars & black holes.
No one asked, but: Freeman Dyson pointed out [0] some astounding properties of a short-period white dwarf binary. (Particularly: there are two stars; both are extremely dense; and they are orbiting around each other at an extremely high relative velocity). If you could perform a close slingshot maneuver around one of these stars, you would be accelerated to relativistic speeds in mere seconds. What's more, this would be an entirely unpowered, propellantless manuevuer (stealing orbital momentum from the binary system). And finally, as an incredible example of the equivalence principle, human passengers wouldn't even notice the insane ~10^4 g's of acceleration: it's equivalent to free-fall.
[0] This is a PDF https://www.ifa.hawaii.edu/~barnes/ast242_s14/Dyson_Machines... (1963)
Not a physicist, but it seems like you should be able to do something similar to aerobraking (not one-to-one since the Sun's "atmosphere" is different from a planet's). Maybe using a solar sail of some sort directed at the right angle?
If I repeat this long enough, my engine will melt, but that's a minor matter of materials science. Maybe I can win my drag race before it melts. Is that the limiting factor or is there some other reason I'm missing that we need a heat gradient?
If you do draw it with enough detail, you'll notice that if you do not heat it more at each turn, you won't be able to get any pressure difference at your "explosion".
If you can keep your cool, why not? Matter is matter.
What's better, you could technically radiation-brake around a star. I don't know how big of a solar sail you'd have to present to get a meaningful force from it.
You can only slingshot around the sun if you were already in a hyperbolic orbit w.r.t the sun. An object in a closed helio centric orbit cannot gain any momentum by "slingshotting" around the sun.
Can you aerobrake around a star?