The answer is "yes", and the modifications are relatively straightforward for experts to make (which is why you may not see it often).
There are roughly three classes of modifications, I'll try to briefly mention all of them.
Throughout, I will be referring to the standard "Regev-type" encryption scheme
$$\mathsf{Enc}_s(m) = (A, As + e + (q/2)m),\qquad \mathsf{Dec}(A, b) = \lfloor b - As\rceil_2$$
where the function $\lfloor x\rceil_p = p\lfloor x/p\rceil$ rounds $x$ to the nearest integer multiple of $p$ (and $\lfloor x\rceil$ is the standard "round to the nearest integer" function).
First, there is a standard way to go from a message space of $\mathbb{Z}/2\mathbb{Z}$ to $(\mathbb{Z}/2\mathbb{Z})^n$, namely via a "matrix encryption scheme".
The idea is to have $n$ independent keys $s_1, s_2,\dots, s_n$.
You can collect these into a matrix $\mathbf{S} = [s_1,\dots,s_n]$, and then encryption with
$$\mathsf{Enc}_{\mathbf{S}}(\vec m) = (A, A\mathbf{S} + \vec e + (q/2)\vec m)$$
Practically, we are "reusing" $A$ across $n$ different encryptions. As $A$ is the largest (size wise) part of the scheme, this is a decent savings. Keys increase by a multiplicative factor $n$ though. I believe this optimization is mentioned in PVW08 ("Lossy Trapdoor Functions and their Applications" maybe?), but don't know if this was the first occurrence of it.
Another way to go from a message space of $\mathbb{Z}/2\mathbb{Z}$ to $(\mathbb{Z}/2\mathbb{Z})^n$ is to use more general rings, i.e. use RLWE. This is somewhat mathematically non-trivial, so I'll give a high-level overview.
Ciphertexts are now of the form $(a, as + e + (q/2)m)$, where now $a, s, e, m$ are all polynomials of degree $n$.
In particular, one gets encryption for bit vectors in $(\mathbb{Z}/2\mathbb{Z})^n$ "for free", in particular without having to increase the size of the secret key. This is one of the more performant ways to go above bit encryption, and is incredibly popular in practice (for example, every NIST solution uses some version of this, i.e. either RLWE, MLWE, or non-integer NTRU stuff, with the exception of FrodoKEM, which intentionally doesn't for security reasons).
What if you don't like the appearance of $\mathbb{Z}/2\mathbb{Z}$ everywhere?
The stories above can all be generalized to have message space $\mathbb{Z}p/\mathbb{Z}$ (rather than the specific case of $p = 2$) by switching the term $(q/2)m$ to $(q/p)m$ (where in the first case, we choose $q$ such that $2\mid q$.
After generalizing, we want $p\mid q$).
This yields encryption with message space $\mathbb{Z}/p\mathbb{Z}$, or after the two optimizations I mentioned $(\mathbb{Z}/p\mathbb{Z})^n$.
Note that this doesn't come for free --- very roughly speaking, the term $(q/2)m$ is used to ensure correct decryption holds, and works in the presence of error $|e| < q/4$ in each coordinate.
For general $p$, this bound tightens, and one must have error $|e| < q/2p$ in each coordinate.
This smaller error leads to less secure schemes.
One can get around this via reparametrizing things, but the point is that switching from $\mathbb{Z}/2\mathbb{Z}\mapsto \mathbb{Z}/p\mathbb{Z}$ doesn't come "for free".
For your updated question, it is worth mentioning that there are LWE-based encryption schemes (even FHE!!) with ciphertext expansion factor (asymptotically optimal), see
