After reading @DocBrown 's remark that the Wikipedia page on IDE's has a History section with the answer, and @YannisRizos 's objection that the first IDE listed there is not a sourced statement, I scrolled down one further paragraph in that Wikipedia page and read:
Maestro I is a product from Softlab Munich and was the world's first integrated development environment 1975 for software.
where  is a bibliography source (in German): "Interaktives Programmieren als Systems-Schlager". The article is from Computerwoche, a magazine directed at IT managers and CIOs, originally published September 21, 1975. They describe the Maestro I system as a 'Programm-Entwicklungs-Terminal-System', which translates as 'program development terminal system'.
A 'similar system' is mentioned: 'Stico' from a company called 'ZEDA', which was around a year and half before August 1976, which makes it around February 1975. However, STICO is described in the second linked article as a 'system that creates out of few instructions a complete program and converts it immediately into an executable object program', which looks more like an alternative language, or a set of macros, or a compiler, than an IDE.
In trying to understand which features of Maestro I would classify it as an IDE, it seems the main innovation at the time was the interactive development it allowed:
Maestro fed each keystroke directly to the CPU producing immediate feedback. This feedback was also enabled by the particular characteristics of the hardware, specifically the use of a keyboard and console instead of the earlier punchcards or tape.
which makes me wonder what is the difference to a LISP REPL, then. But hey, I just wanted to overcome the 'cite your sources' problem mentioned above.