Yes, as a planet spins faster and faster, then first it becomes an oblate spheroid like this

(that's exaggerated compared to our planets though - none of our planets are flattened as much as this)

That's approximately what happens with Earth - it's somewhat flattened at its poles, though only by 21 kilometers, see Reference ellipsoid.

But as it rotates faster and faster, eventually the prolate spheroid, or more generally, a triaxial ellipsoid is preferred. Jacobi proved that surprising result in 1841.

The mathematician Jacobi, who predicted that rapidly spinning planets could take the form of triaxial ellipsoids in 1834

So, if a planet rotates very quickly, it will look this - this is an artist's impression of a dwarf planet that was found in the Kuiper belt,around 50 times further from the sun than Earth,  in 2004:

Haumea: Rugby Ball Planet - artist's impression

This is the shape of a triaxial ellipsoid:

Here is an artist's rendition of Haumea rotating showing its distinctive red spot:

Haumea: Rugby Ball Planet

It has a day only four hours long, and is 1,920 × 1,540 × 990 km

Then, there's another possibility too. Though we don't know any planets like this, there are many contact binary asteroids and comets, which are roughly dumbell shaped.  For instance comet 67p which Rosetta and Philae visited.

They don't have enough gravity to be rounded like a planet. But you could have the same for planets. You could in principle even have an Earth sized planet almost touching our Earth - even with a shared atmosphere and ocean.

(Image NASA)  'Double Earths' Could Be Fun Exoplanets To Hunt For -- If They Existsee: Can binary terrestrial planets exist?

This is explored fictionally in Robert Forward's "Rocheworld" which is an "overcontact binary"

Rocheworld

And in principle a planet spinning at just the right rate could have three, or four lobes, and it could also be donut shaped. Nothing has yet been observed resembling this, but theoretically it's possible.

For more on this: How and why are planets spherical? What makes them round?

I wrote it up for my Science20 blog as So You Thought You Knew What Spinning Planets Look Like? ... Surprising Shapes Of Rapidly Spinning Planets

It's also one of the sections in my kindle book

Simple Questions - Surprising Answers - In Astronomy, Robert Walker - Amazon.com