If Archimedean local fields are ok, then the simplest example probably occurs with $G=GL(2, \mathbb R)$ and $K=SO(2, \mathbb R).$ The irreducible representations of $K$ are in bijection with the integers. One can construct representations of $GL(2, \mathbb R)$ such that the set of $K$-types is the set of odd integers, the set of even integers, the set of odd integers with absolute value at least (or at most) $2n+1,$ or the set of even integers with absolute value at least (or at most) $2n.$ In particular, one can construct two representations such that the restrictions to $K$ differ by deleting a single representation of $K.$

In a bit more detail, take $s=(s_1, s_2)$ a pair of complex numbers and $\omega=(\omega_1, \omega_2)$ where for $i=1,2,$ $\omega_i$ is either trivial or the sign character. Consider smooth functions $GL(2, \mathbb R) \to \mathbb C$ such that
$$
f\left( \begin{pmatrix} a& b \\ 0& d \end{pmatrix} g
\right)
= |a|^{s_1} \omega_1(a) |d|^{s_2} \omega_2(d) f(g).
$$
The $K$-type corresponding to an integer $n$ will appear in this representation if and only if $(-1)^n = \omega_1(-1)\omega_2(-1).$ (And all have multiplicity one.) You get reducibility when $s_1-s_2$ is an integer $m$ with this parity. If $m$ is positive there is a subrepresentation which lives on the $K$-types $\ge m$ in absolute value. If $m$ is nonpositive, there is a subrepresentation which lives on the $K$-types $\le -m$ in absolute value.

For obvious reasons, the reference I know best is the book Goldfeld and I wrote. We discuss reducibility of $(\mathfrak{g}, K)$-modules on p. 267. On p. 332, we give a fairly elementary description of each invariant subspace looks like in the representation of $GL(2, \mathbb R)$ on smooth functions. I see another reference in the comments as well.

Adding: in the $p$-adic case a similar phenomenon occurs and may be easier to see. Take $F$ a nonarchimedean local field with ring of integers $\mathfrak{o}.$ Take $G=GL(2, F)$ and $K=GL(2, \mathfrak{o}).$ Consider the representation $Ind_B^G \chi$ parabolically induced from a character of the Borel subgroup $B$:
$$
\chi\begin{pmatrix} a& b \\ 0& d \end{pmatrix} = |a|^{s_1}|d|^{s_2}.
$$
As a $K$-module $Ind_B^G \chi$ is isomorphic to $Ind_{B\cap K}^K 1$ for all $s_1, s_2.$ (Here "$1$" means the one dimensional trivial representation of $B \cap K$.) As a $G$-module $Ind_B^G \chi$ is irreducible for most values of $s_1, s_2,$ but if $s_1=s_2$ then the subspace of $Ind_{B\cap K}^K 1$ corresponding to the trivial representation of $K$ is actually a $G$-invariant subspace spanned by the function $g \mapsto |\det g|^{s_1}.$