  Let $p>3$ be a prime. We show that, for each integer $d$ with $p \leq d \leq
2(p-1)$, there exists a generalized almost perfect nonlinear (GAPN) binomial or
trinomial over $\mathbb{F}_{p^2}$ of algebraic degree $d$. We start by deriving
sufficient conditions for the function $G \colon \mathbb{F}_{p^2} \rightarrow
\mathbb{F}_{p^2}, X \mapsto X^{d_1} + u X^{d_2}$ to be GAPN in the case where
one of the terms of $G$ is GAPN. We then give explicit constructions of GAPN
binomials over $\mathbb{F}_{p^2}$ of any odd algebraic degree between $p$ and
$2(p-1)$ and, in the case where $p$ is not a Mersenne prime, also of any even
algebraic degree in this range. To obtain GAPN functions of even algebraic
degree also in the general case, we finally show how to construct GAPN
trinomials over $\mathbb{F}_{p^2}$ of any even algebraic degree between $p$ and
$2(p-1)$ by applying a characterization of a special form of GAPN binomials by
\"{O}zbudak and S\u{a}l\u{a}gean. Our constructed functions are the first GAPN
functions of even algebraic degree over extension fields of odd characteristic
reported so far.
