Assembly language or assembly code is the most typical usage. There's nothing wrong with calling it assembler code either, but it's less common.
Source: I started writing assembly code in 1968, over the years coding for Sigma 5/7, PDP-10, TI 9900, 6502, 8080, Z80, 8086, 68000, x86, and ARM. I occasionally saw it called assembler language, but less often than assembly language.
Update: As I think about it, there is a bit more nuance to the terminology that I remembered at first. While I do recall "assembler language" and "assembler code" as being infrequently used, "assembler" by itself has often been a common usage:
"I'm writing this in assembly code."
"I'm writing this in assembler."
Those were used fairly interchangeably, even by someone like me to whom "assembler code" sounded a bit off.
So you're definitely not wrong to call it "assembler". Where you're going wrong is claiming that "assembly code" is incorrect.
Source: I started writing assembly code in 1968, over the years coding for Sigma 5/7, PDP-10, TI 9900, 6502, 8080, Z80, 8086, 68000, x86, and ARM. I occasionally saw it called assembler language, but less often than assembly language.
Update: As I think about it, there is a bit more nuance to the terminology that I remembered at first. While I do recall "assembler language" and "assembler code" as being infrequently used, "assembler" by itself has often been a common usage:
"I'm writing this in assembly code."
"I'm writing this in assembler."
Those were used fairly interchangeably, even by someone like me to whom "assembler code" sounded a bit off.
So you're definitely not wrong to call it "assembler". Where you're going wrong is claiming that "assembly code" is incorrect.