Here is a novel, explicit answer to your question :
Benedict Gross and Mark Reeder have recently discovered a family of supercuspidal representations, called "simple supercuspidal representations", of simply connected, split, almost simple, reductive groups, and they fit your question (i.e. they show that the answer to your question is yes) These representations are examples of what you are looking for.
Here's an example of a simple supercuspidal representation, and the general case is similar. Consider $G(\mathbb{Q}_2) = SL(2,\mathbb{Q}_2)$. Let $I$ be an Iwahori subgroup of $SL(2,\mathbb{Q}_2)$, and let $I^+$ be the pro-unipotent radical of $I$. For example, we can take
$$I = \begin{bmatrix} \mathfrak{o} & \mathfrak{o}\\\ \mathfrak{p} & \mathfrak{o} \end{bmatrix}$$
$$I^+ = \begin{bmatrix} 1 + \mathfrak{p} & \mathfrak{o}\\\ \mathfrak{p} & 1 + \mathfrak{p} \end{bmatrix}$$
Let $\chi$ be an "affine generic character" of $I^+$. Since we are dealing with $SL(2,\mathbb{Q}_2)$, this just means the following : Let $\eta$ be the character of $\mathbb{Z}_2$ given by $$\eta : \mathbb{Z}_2 \rightarrow \mathbb{C}^*$$ $$2 \mathbb{Z}_2 \mapsto 1$$ $$1 + 2 \mathbb{Z}_2 \mapsto -1$$ Then define the character
$$\chi : I^+ \rightarrow \mathbb{C}^*$$ $$\begin{bmatrix} a & b\\\ 2c & d \end{bmatrix} \mapsto \eta(b) \eta(c)$$
Then $Ind_{I^+}^{SL(2,\mathbb{Q}_2)} \chi$ is a supercuspidal representation (which Gross and Reeder call a "simple supercuspidal representation", since it is so "simple" to define). The general situation is similar : If $G$ is simply connected, split, almost simple, $F$ is a $p$-adic field, $I \subset G(F)$ is an Iwahori subgroup, $I^+ \subset I$ is the pro-unipotent radical, and $\chi : Z I^+ \rightarrow \mathbb{C}^*$ is an "affine generic character" (where $Z$ is the center of $G(F)$), then $Ind_{Z I^+}^{G(F)} \chi$ is a supercuspidal representation (called a "simple supercuspidal representation"). For more details, check out Section 9 of https://www2.bc.edu/~reederma/AdjointGamma.pdf