Did Nintendo Release a 32-Bit Handheld System?

No. Nintendo never released a dedicated 32-bit handheld console. The closest hardware is the Game Boy Advance (2001), often categorized in the "32-bit era" but technically using a 16-bit CPU (ARM7TDMI) with a 32-bit system bus.

The 32-bit generation was primarily defined by home consoles: Nintendo 64 (64-bit), Sony PlayStation (32-bit), and Sega Saturn (32-bit). Nintendo's handheld line remained 16-bit (GBA) until the Nintendo DS (2004), which used dual ARM processors (32-bit and 16-bit) but is considered a separate generation.