Immersions of the hyperbolic plane

Yes, it immerses isometrically into certain solvmanifolds. Take an Anosov map of $T^2$, such as $\left[\begin{array}{cc}2 & 1 \\1 & 1\end{array}\right]$. The mapping torus admits a locally homogeneous metric modeled on the 3-dimensional unimodular solvable Lie group. The matrix has two eigenspaces with eigenvalues $\frac{3\pm\sqrt{5}}{2}$, and the suspensions of lines on the torus parallel to these eigenspaces give immersed manifolds modeled on $\mathbb{H}^2$. If the eigenspace line contains a periodic point of the Anosov map, the mapping torus of it will be an annulus. But otherwise it will be an immersed injective totally geodesic hyperbolic plane, which I think is what you're asking for.

Here is a general construction. Take a non-trivial representation of $H=\text{SL}_2(\mathbb{R})$ into a semisimple Lie group $G$, take $K<G$ a maximal compact subgroup and take $\Gamma<G$ an irreducible cocompact lattice. Endow $X=G/K$ with the standard symmetric space structure and consider the image of $H$ in $X$ which is a totally geodesic hyperbolic space. Its image in $\Gamma\backslash X$ will be a totally geodesic immersion of a hyperbolic plane into a compact Riemannian manifold.

Further, if $H$ is not a factor of $G$, up to Baire generically conjugating $\Gamma$ in $G$, we can get that the image of $H$ will be non-compact and if $X$ is of dimension $\geq 5$ (e.g $G=\text{SO}(5,1)$ or $G=\text{SL}_3(\mathbb{R})$) we can get that the immersion is injective (thanks to Ian Agol for correcting an inaccuracy here in a previous version of my answer).

This is an attempt to visualize the answer by Ian Agol. I am not sure it is correct. If it is, it must be a fundamental domain for the group action from that answer, and the surfaces - totally geodesic images of various projective plane embeddings.