OK Many thanks for your patience. I did a test this morning and will try to attach the three photos here. On each of the three shots the lightening bolt symbol in the bottom right corner of the viewfinder was flashing. (I took other shots where it was not flashing in the same area also) In the Custom Settings Menu the Exposure Delay Mode is set to off.
I do try to see and remember the settings as I take a photo but they are only in the viewfinder for seconds and I am sure I would forget them before I had time to write anything down
1) Exposure 1/400 - f/10 - ISO800
2) Exposure 1/400 - f/10 - ISO800
3) Exposure 1/500 - f/11 - ISO800
This is no doubt very frustrating for everyone.

There is no reasonable explanation why the Flash-Ready indicator would blink in bright sun when the flash was not being used. There is even less reason why it would do this in P mode but not in A mode. No one will know why this time, we simply need more info about the situation. Aperture would be about the only candidate, but Auto ISO could affect it. As is, we know nothing. Frankly, it is sounding like a camera defect. Certainly it is not normal operation.
An experienced troubleshooter with access to the camera (like a repair shop) would try different things, hoping to chance onto any additional information that might show some possible cause.
Short of that access, it would at least be be good to see the Exif data (the complete data).
The camera records in the image file just about every possible setting that was used. This is called Exif data, and most photo programs show it, or at least some of it. However, when Adobe programs rewrite the file, some of the Nikon data is omitted then. And worse, when Adobe programs save it with the File - Save For Web menu, they omit ALL of the Exif data.
Also, even if present in your image upload, some posting host sites strip away the Exif data. I am not sure about this site, but it certainly does appears to strip it, at least I have never seen an exception here with the Exif data. So sorry, I doubt posting the image here can help. If you could post it on your own web site, the Exif probably would show.
So... it's very difficult to know more than what you can tell us.
You did post guesses about exposure, but frankly, if ISO 800, Sunny 16 says f/11 would be about 1/1600 second shutter in bright sun. That would be the normal expectation. But you said 1/400, and that does not compute. There are some darker areas in your pictures, but the bright areas mostly do not look overexposed.
Also the big key would would see data for BOTH the P mode case when it blinked, and also see the A mode case when it did not, to try to find out what the difference was. There must be some difference. In camera A mode, you set aperture yourself, but we have no clue what it was. Seeing the Exif data can only help to know what the situation was.
Here is one way to see more, if you care to proceed. Tedious to describe, but easy to do.
You could go to this site:
PhotoME - Exif, IPTC & ICC Metadata Editor
and download their free Exif viewer Photome. It is one of the more excellent ones, and it is free.
Then you could install it, and use its File - Open menu to open an original image straight from the camera (JPG or RAW, it does not matter, but one that does this problem). However, if you have let Adobe resave it, much Exif stuff will be gone then, only the basics left.
Then you could select its menu File - Export, as Text Output.
Then to the right side, you could select Sections checked to be Image, Camera, Manufacturer note (all three).
You can see it then. We could see it if you continue this way:
Then you click at the bottom: Copy to Clipboard.
Then you can compose a reply here, and position the mouse cursor where you want insert this, and hold CTRL and type V, which will Paste the clipboard text into that message. There is a lot of it (settings from this picture):
Here is an arbitrary sample of the Exif data that you would see then (and we could see then):
But you can see that there is much more you could tell us.
[ PhotoME ]
PhotoME version: 0.8ß2 (Build 891)
[ Overview ]
File name: E:\Camera\20130510\800_4059.NEF
File type: Nikon Camera RAW
File size: 32,903 KB
Creation date: 5/10/2013 15:26
Last modification: 5/10/2013 15:26
Make:
NIKON CORPORATION
Camera: NIKON D800
Lens: AF-S Zoom-Nikkor 24-70mm F2.8G ED
Software: Ver.1.02
Focal length: 38 mm (equiv. 38 mm)
Aperture: F5.6
Exposure time: 1/60"
ISO speed rating: 100/21°
Program: Aperture priority
Metering Mode: Pattern
White Balance: Flash
Focus Mode: AF-S
Image Stabilizer: Off
Noise Reduction: Off
Flash: Flash fired, compulsory flash mode, return light detected
[ Image ]
New subfile type: Reduced-resolution image data
Image width: 160 px
Image height: 120 px
Number of bits per component: 8, 8, 8
Compression scheme: uncompressed
Pixel scheme: RGB
Manufacturer: NIKON CORPORATION
Image input equipment model: NIKON D800
Image data location: 0x0001F7D0
Orientation of image: 0° (top/left)
Number of components: 3
Number of rows per strip: 120 rows
Bytes per compressed strip: 57600 bytes
Image resolution in width direction: 300 dpi
Image resolution in height direction: 300 dpi
Image data arrangement: Chunky Format (Interleaved)
Unit of X and Y resolution: inch
Software: Ver.1.02
File change date and time: 2013-05-10 15:26:19
Person who created the image:
SubIFD Pointer: 0x0002D8D0, 0x0002D948, 0x0002DA2C
Pair of black and white reference values: [0, 255, 0, 255, 0, 255]
Copyright holder:
Exif IFD Pointer: 0x00000668
GPS IFD Pointer: 0x0001F7BC
Date and time of original data generation: 2013-05-10 15:26:19
TIFF/EP Standard ID: 1.0.0.0
[ Camera ]
Exposure time: 1/60"
F number: F5.6
Exposure program: Aperture priority
ISO speed rating: 100/21°
??? (8830): 2
Date and time of original data generation: 2013-05-10 15:26:19
Date and time of digital data generation: 2013-05-10 15:26:19
Exposure bias: ±0 EV
Maximum lens aperture: 3 Av (F2.8)
Metering mode: Pattern
Light source: Flash
Flash: Flash fired, compulsory flash mode, return light detected
Lens focal length: 38 mm
Manufacturer notes: 0x00000888
User comment:
DateTime subseconds: 0.6"
DateTimeOriginal subseconds: 0.6"
DateTimeDigitized subseconds: 0.6"
Sensing method: One-chip color area sensor
File source: Digital Camera
Scene type: A directly photographed image
CFA pattern: [Red, Green], [Green, Blue]
Custom image processing: Normal process
Exposure mode: Auto exposure
White balance: Manual
Digital zoom ratio: Digital zoom was not used
Focal length in 35 mm film: 38 mm
Scene capture type: Standard
Gain control: None
Contrast: Normal
Saturation: Normal
Sharpness: Normal
Subject distance range: Unknown
[ Manufacturer notes ]
MakerNote Version: Version 2.1.0
ISO: 100/21°
Quality: RAW
White Balance: Flash
Focus Mode: AF-S
Flash Setting: Normal
Flash Type: Optional,TTL-BL
White Balance Fine Tune: ±0, ±0?
Color Balance 1: 2.1875, 1.20703125, 1, 1
Program Shift: ±0 EV
Exposure Difference: -2.17 EV
Preview IFD Pointer: 0x00005A06
Flash Exposure Compensation: ±0 EV
ISO Setting: 100/21°
??? (0017): 1.6.0
Flash Exposure Bracket Value: ±0 EV
Exposure Bracket Value: ±0 EV
Hi-Speed Crop: Off (7424x4924 cropped to 7424x4924 at pixel 0, 0)
??? (001C): 1.6
Serial Number: 3046677
Color Space: sRGB
VR Info Version: 1.0.0
Vibration Reduction: Off
??? (001F): 2 (Off?)
Active D-Lighting: Off
Picture Control Version: 1.0.0
Picture Control Name: STANDARD
Picture Control Base: STANDARD
Picture Control Adjust: Default Settings
Picture Control Quick Adjust: ±0
Sharpness: +3
Contrast: ±0
Brightness: ±0
Saturation: ±0
Hue Adjustment: ±0
Filter Effect: not set
Toning Effect: not set
Toning Saturation: Not set
Time Zone: UTC -6:00 (Chicago, Houston, Mexico City)
Daylight Saving Time: Off
Date Format: Month/day/year
ISO: 100/21°
ISO Expansion: Off
ISO (2): 100/21°
ISO Expansion (2): Off
Vignette Control: Normal
Lens Type: D, G
Lens: 24-70mm F2.8
Flash Mode: Fired, External
Shooting Mode: Single-Frame
Auto Bracket Release: Auto Release
Lens F-Stops: 6
Shot Info Version: 2.2.2
RAW Compression: Lossless
Noise Reduction: Off
Color Balance Version: 2.1.7
Lens Data Version: 2.0.4
Used Lens: AF-S Zoom-Nikkor 24-70mm F2.8G ED
Exit Pupil Position: 107.8 mm
AF Aperture: F2.9
??? (0098): 9
??? (0098): 3
Focus Position: 0x42
??? (0098): 64
Focus Distance: 2 m
Focal Length: 38.9 mm
Lens ID Number: 147
Lens F-Stops: 6
Min Focal Length: 24.5 mm
Max Focal Length: 71.3 mm
Max Aperture At Min Focal: F2.8
Max Aperture At Max Focal: F2.8
MCU Version: 149
Effective Max Aperture: F2.8
??? (0098): 3
??? (0098): 72
??? (0098): 26
??? (0098): 82
??? (0098): 6
??? (0098): 0
??? (0098): 0
??? (0098): 1
??? (0098): 2
??? (0098): 0
??? (0098): 0
??? (0098): 0
??? (0098): F1.1?
RAW Image Center: x = 3712, y = 2462
Retouch History: None
??? (00A3): 0
??? (00A4): 3.0.0
Shutter Count: 4278
Multi Exposure Data Version: 1.0.0
Multi Exposure Mode: Off
Multi Exposure Shots: 0
Multi Exposure Auto Gain: Off
High ISO Noise Reduction: Normal
AF Info Version: 1.0.0
??? (00B7): 0
AF Area Mode: Single-point AF
??? (00B7): 0x0
Auto Focus: On
??? (00B7): 1
File Informations Version: 1.0.0
Directory Number: 100
File Number: 4059
??? (00B9): 0x00FF0000