Are there any Baryons that have quark-antiquark combinations?

No, a three-quark baryon can not be be made out of two quarks and one anti-quark (and vice versa) as this would necessarily give the particle color.

Each quark carries one of three colors (red, blue, green) and each anti-quark respectively carries anti-color. Color is an additive quantity when constructing particle and the result must be color-neutral, i.e. it either is made out of for example red + anti-red or red + blue + green quarks. Although we can not observe the "color" of the constituents of a particle directly (as all observable particles must be color-neutral), we can measure its effects indirectly via certain cross-sections.

With this being said, we can now clearly see that there is no way in which we can construct particles of two-quarks and one anti-quark as any possible combination would not be color-neutral. Hence, color-neutrality forbids observable three-quark particles composed out of quarks and anti-quarks.

Penta-quarks however can and must contain quarks and anti-quarks as mentioned in a previous answer.

Pentaquarks contain three quarks and a quark-antiquark pair, and they are baryons, since baryons are defined as having an odd number of valence quarks.

Yes, there are some oddball baryons that contain four quarks and one anti-quark. They're called pentaquarks. These could be thought of as a baryon with three quarks plus a meson with a quark anti-quark pair, but sticking to each other more than they should. There are few hadrons that survive more than a tiny fraction of a second, and only the proton and neutron live longer than one second. The "sticky" baryon-meson combinations are found through analysis of scattering data at the major particle accelerators.

The known pentaquarks are: $P_c(4450)^+$, $P_c(4380)^+$ ([CERN 2015][1], [Arxiv paper][2]) and $P_c(4312)^+$ ([CERN 2019][3]) - oh but wait, the 4450 one seems to really be two pentquark peaks close together in mass - 4440 and 4457.[CERN][4] BTW, the subscript 'c' indicates that one of the constituent quarks is a c (charmed) quark.

There's a tetraquark - two quarks and two anti-quarks in one bag - with mass 4430, named $Z(4430)^+$ (nothing to do with the Z boson). [CERN][5] Can it be described as two mesons being "sticky" with each other or is it better described as a more tightly-knit group of four of the little buggers?

Another possible exotic hadron is made of six quarks all stuck together. This is a "dibaryon". When I was an undergrad in physics, I worked with Dr. Yokosawa at Argonne National Laboratory on finding bumps in polarized proton-proton scattering. That was an interesting experience. As of last time I checked the literature, there was [only one found][6] that stood up to analysis, but always there are candidates due to unexplained wiggles in cross-section vs energy plots. These usually turn out to be something less exotic. Interesting, but since anti-quarks aren't involved, never mind!

[1]: CERN’s LHCb experiment reports observation of exotic pentaquark particles (14 JULY, 2015)

[2]: Observation of J/ψp resonances consistent with pentaquark states in Λ0b→J/ψK−p decays (LHCb collaboration)

[3]: LHCb experiment discovers a new pentaquark

[4]: 26 March 2019: Observation of new pentaquarks

[5]: 9 April 2014: Unambiguous observation of an exotic particle which cannot be classified within the traditional quark model.

[6]: On the History of Dibaryons and their Final Observation, H. Clement